truncated cube

The truncation procedure that produced the truncated cube I can be applied to arbitrary polyhedra so as to obtain new polyhedra.

2003, Saul Stahl, Geometry from Euclid to Knots, published 2010, page 278:

The truncated cubes are obtained by symmetrically edging the eight vertices of the cube by l × t, where l is the length of the cubes side and 0 < t ≤ 1/2. When t = 1/2, a cuboctahedron is obtained.

2012, Cecilia Noguez, Ana L. González, “11: Localized Surface Plasmons of Multifaceted Metal Nanoparticles”, in Tapan K. Sau, Andrey L. Rogach, editors, Complex-shaped Metal Nanoparticles: Bottom-Up Syntheses and Applications, page 384:

In this subsection the optical behavior of different truncated cubes will be analyzed and compared to those of a cube and of a sphere of the same dimensions.

2013, S. D'Agostino, “Chapter 3: Size and Shape Dependence of Localized Surface Plasmon Resonances”, in Fabio Della Sala, Stefania D'Agostino, editors, Handbook of Molecular Plasmonics, page 152:

Truncated cubes can be packed sharing their octagonal faces, the remaining voids are filled by octahedra (Figure 10(b)).

2014, Walter Steurer, “1: Crystal Structures of Metallic Elements and Compounds”, in David E. Laughlin, Kazuhiro Hono, editors, Physical Metallurgy, page 20:

(geometry) A polyhedron with eight triangular and six octagonal faces that is an Archimedean solid and can be constructed by truncating a cube at each vertex along each adjoining edge.


snub dodecahedron

en-noun

snub dodecahedron (plural snub dodecahedra or snub dodecahedrons)

snub dodecahedra (plural)
snub dodecahedrons (plural)

From dodecahedron simum, Kepler's name for this solid in is 1619 work Harmonices Mundi.

This gives two points, A’, A₁’, which are projections of vertices of the two enantiomorphous snub dodecahedra that can be inscribed in the regular dodecahedron.

1935, L. Lines, Solid Geometry, page 182:

The form can be derived by triangulating the pentagonal faces of the snub dodecahedron (Fig. 19).

1992, Gábor Gévay, “Icosahedral Morphology”, in István Hargittai, editor, Fivefold Symmetry, page 198:

In the case of the latter, two chiral members, the snub cube and the snub dodecahedron, are realized.

2001, Ehud Keinan, Israel Schechter, Chemistry for the 21st Century, page 138:

The snub dodecahedron has 60 vertices, 150 edges and 92 faces of which 12 are pentagonal and 80 triangular. This polyhedron has I symmetry and can therefore be constructed in left- and right-handed forms.

2006, P. W. Fowler, D. E. Manolopoulos, An Atlas of Fullerenes, page 50:

The result is a structure of two nested snub cubes rotated by an angle of 45° against each other (n = 24) and two nested snub dodecahedra, where the rotation angle amounts to 36° (n = 60).

2010, Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 51–8:

(geometry) A polyhedron that has 12 pentagonal and 80 triangular faces and is an Archimedean solid.

snub icosidodecahedron

polyhedron with 12 pentagonal and 80 triangular faces


great rhombicuboctahedron

great rhombicuboctahedron (plural great rhombicuboctahedra or great rhombicuboctahedrons)

great rhombicuboctahedra (plural)
great rhombicuboctahedrons (plural)

From great + rhombicuboctahedron, referring to the fact that, similarly to the rhombicuboctahedron, the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron, the dual of the cuboctahedron.

Figure 244 — the structure for this complicated design is the great rhombicuboctahedron, which consists of octagonal, hexagonal, and square faces.

1977, Wucius Wong, Principles of Three Dimensional Design, page 70:

Zeolite A consists of truncated octahedra flanking the edges of a larger cube; they are connected along the edges by smaller cubes, and the truncated octahedra and cubes flank a great rhombicuboctahedron at the center (28).

2005, M. Cahay, Nanoscale Devices, Materials, and Biological Systems: Fundamentals and Applications : Proceedings of the International Symposium, page 513:

The great rhombicuboctahedron can be made by truncating the cuboctahedron's vertices.

2012, Edward S. Popko, Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, page 169:

(geometry) An Archimedean solid that is isogonal and has twenty-six regular faces (twelve square, eight hexagonal and six octagonal).

cubottaedro troncato

polyhedron with 12 square, 8 hexagonal and 6 octagonal faces


truncated cuboctahedron

truncated cuboctahedron (plural truncated cuboctahedra or truncated cuboctahedrons)

truncated cuboctahedra (plural)
truncated cuboctahedrons (plural)

From truncated + cuboctahedron, referring to a partial construction (the resultant figure needs further distortion to make the faces regular).

The H'(S-S) type of linked truncated octahedra and truncated cuboctahedra is analogous to the fluorite structure with the truncated octahedra in the positions of the fluorine atoms and the truncated cuboctahedra in the positions of the calcium atoms.

1962, The Mineralogical Magazine and Journal of the Mineralogical Society, volume 33, page 1013:

A three-dimensional network of linked cavities, each with a truncated cuboctahedron shape (Figure 6.5) through the structure, is thus formed; the truncated cuboctahedra are also space-filling (each shares its octagonal face with six others), forming channels that run parallel to the three axial cubic directions through these large cavities.

2012, Lesley E. Smart, Elaine A. Moore, Solid State Chemistry: An Introduction, 4th edition, page 274:

Its pore structure consists of truncated octahedra linked to other cavities through 6-membered rings and of truncated cuboctahedra linked together through 8-membered rings.

2013, István Palinkó, Zoltán Kónya, Ákos Kukovecz, Imre Kiricsi, “22: Zeolites”, in Robert Vajtai, editor, Springer Handbook of Nanomaterials, page 822:

great rhombicuboctahedron
great rhombicuboctahedron

(geometry) An Archimedean solid that is isogonal and has twenty-six regular faces (twelve square, eight hexagonal and six octagonal); a great rhombicuboctahedron.


oscnode

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Three nodes may coincide as consecutive points of a curve of finite curvature, giving rise, not to a triple point, but to the singularity called an oscnode; this is, in fact, an osculation or three-pointic contact of two branches of the curve.

1879, George Salmon, A Treatise on the Higher Plane Curves, page 214:

Rhamphoid cuspidal lines possess two kinds of singular points which affect the class of the surface, and occur (i) when the rhamphoid cusp changes into an oscnode, (ii) when there are cubic nodes on the surface.

1908, A. B. Basset, “Singular liens and curves on surfaces”, in The Quarterly Journal of Pure and Applied Mathematics, volume 39, page 338:

A tacnode is defined to be a singularity equivalent to the origin in the curve y²-x²#61;0, that is, the union of two smooth arcs meeting with contact of order two. Similarly, an oscnode is a singularity equivalent to y²-x⁶#61;0, i.e., consisting of two smooth branched with contact of order three, and so on.

2012, Joe Harris, Algebraic Geometry: A First Course, page 262:

(geometry) A node of a plane curve where two branches meet with contact order 3.


truncated dodecahedron

truncated dodecahedron (plural truncated dodecahedra or truncated dodecahedrons)

truncated dodecahedra (plural)
truncated dodecahedrons (plural)

From truncated + dodecahedron, referring to a construction method for the figure.

1813, Spinell, entry in John Mason Good, Olinthus Gregory, Newton Bosworth (editors), Pantologia: A New Cyclopaedia, Volume XI: SPA — TZE, unnumbered page,
This, which may be regarded as a brown garnet or schoerl in truncated dodecahedrons of Rome de Lisle, was originally found in the island of Ceylon, among tourmalines and other crystalline substances, with which it was confounded.

The morphologies commonly observed in bcc materials include truncated dodecahedrons with {110} faces and {100} truncations and cubes with {100} faces.

1983, Electron Microscopy Society of America, Proceedings: 41st Annual Meeting, page 232:

Void shapes were found to be generally truncated dodecahedra with {011} faces and {001} truncations.

1990, David S. Gelles, “Neutron Irradiation Damage in Ferritic Fe-Cr Alloys”, in Nicholas H. Packan, R. E. Stoller, A. S. Kumar, editors, Effects of Radiation on Materials: 14th International Symposium, Volume 1, page 90:

1996, T. A. Fournelle, K. W. Weston, Generalizations of Steinberg Groups, World Scientific Publishing, Series in Algebra, Volume 4, page 176,
The edges of the linkages lie on the edges of a truncated dodecahedron. Note that the group of orientation preserving symmetries of a truncated dodecahedron is precisely A₅.

(geometry) An Archimedean solid with thirty-two regular faces (twelve decagons and twenty triangles) and ninety edges.

dodecaedro troncato

polyhedron with 12 decagonal faces, 20 triangular faces and 90 edges


midsphere

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The edges are omitted as irrelevant to the present argument, but the midspheres are still included because Kepler rather casually substitutes the midsphere for the insphere of the octahedron, since it happens to give a value closer to the observed distance.

1999, Richard Padovan, Proportion: Science, Philosophy, Architecture, page 260:

A face of a Catalan solid can be simply constructed by the Dorman-Luke construction [3, 4, 6, 16], which is based on a dualization process known as polar reciprocation [6], with respect to the midsphere.

2003, Rona Gurkewitz, Bennett Arnstein, Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality, page 4:

The Archimedean solids can all be inscribed in a sphere and in one of the Platonic solids. Their duals are the Catalan solids, with faces that are congruent but not regular (face-transitive); indtead of circumspheres like the Archimedean solids, they have inspheres. The midspheres, touching the edges, are common to both of them.

2014, Walter Steurer, “Crystal Structures of Metallic Elements and Compounds”, in David E. Laughlin, Kazuhiro Hono, editors, Physical Metallurgy, 5th edition, volume 1, page 20:

(geometry, solid geometry) The sphere which, if it exists, is tangent to every edge of a given polyhedron and with respect to which the polyhedron's vertices are the inversion poles of the planes of the faces of its dual polyhedron (and vice versa).

intersphere

sphere tangent to every edge of a polyhedron

1994, Tim Gallagher, Bruce Piper, Chapter 7: Convexity Preserving Surface Interpolation, Nickolas S. Sapidis, Designing Fair Curves and Surfaces, page 179,
For any two circles (spheres) there always exists at least one midcircle (midsphere) which inverts the two given circles (spheres) into each other. in the older literature this is also known as the circle (sphere) of antisimilitude. […] The center of a midcircle (midsphere) is the center of similitude of two given circles (spheres). […] Clearly the midcircle (midsphere) of two equal circles (spheres) is the midline (midplane) between the two, which is partial justification for calling inversion in a circle or sphere reflection in a circle or sphere.

(geometry) The sphere, between two given spheres, by which each of the latter may be inverted into the other.

sphere of antisimilitude

intermediate sphere by which each of two given spheres can be inverted into the other


circle of Apollonius

circle of Apollonius (plural circles of Apollonius)

Apollonius of Perga
Circles of Apollonius

Named for the ancient Greek geometer and astronomer Apollonius of Perga (ca 262—ca 190 BCE).

In the present paper an attempt is made to find for the tetrahedron the analogues of the circles of Apollonius of the triangle.

1934, Tôhoku Mathematical Journal, volumes 39-40, page 264:

It follows that O lies on the circle of Apollonius for A and C and ratio OT/OC, and on the circle of Apollonius for B and C and ratio OT/OC. Point O is thus found at the intersections, if any exist, of these two circles of Apollonius. The details are left to the reader.

1995, Howard Whitley Eves, College Geometry, page 165:

More generally, the locus of points z such that #92;left#92;vertx-z#92;right#92;vert#61;#92;alpha#92;left#92;verty-z#92;right#92;vert is a circle of Apollonius which has x and y as inverse points and which cuts each circle through x and y orthogonally.[…]They show loci analogous to circles of Apollonius and the locus of points equidistant from two given points when the norm is an #92;mathcal#123;l#125;#95;p-norm.

1996, A. C. Thompson, Minkowski Geometry, page 126:

(geometry) The figure (a generalised circle) definable as the locus of points P such that, for given points A, B and C, =/(\frac).

(geometry) Any of eight circles (including degenerate cases) that, for a given set of three circles, solve the problem of Apollonius (i.e., intersect each circle tangentially).


generalized circle

generalized circle (plural generalized circles)

We shall speak of lines and circles as generalized circles. A line is a generalized circle that passes through the point at infinity. Under the inversion #92;zeta#61;1#47;z the generalized circle (53.3) goes into
(53-4) β|ζ|²-pζ-◌̅p◌̅ζ+α=0,

1965, H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, page 247

which is a generalized circle in the ζ-plane.

For any Apollonian family of circles defined by the point circles A and B, the Coaxal Circles Theorem states that every generalized circle through A and B meets each of the Apollonian circles at right angles.

1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 252:

This is why it makes sense to call every circle and every straight line a generalized circle. Thus the geodesics of ds consist of all arcs of generalized circles which lie wholly within Ω and are orthogonal to w (see Figure 2).

2001, Elie Zahar, Poincaré's Philosophy: From Conventionalism to Phenomenology, page 84:

(geometry, inversive geometry) A circle or a line, the two being regarded as types of a single form.


Möbius transformation

Möbius transformation (plural Möbius transformations)

Named for German mathematician and theoretical astronomer August Ferdinand Möbius (1790–1868).

2008, J. Vernon Armitage, John Parker, Jørgensen's inequality for Non-Archimedean Metric Spaces, Mikhail Kapranov, Sergii Kolyada, Yu. I. Manin, Pieter Moree, Leonid Potyagailo (editors), Geometry and Dynamics of Groups and Spaces: In Memory of Alexander Reznikov, page 97,
Jørgensen's inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete.

2012, Martin Delacourt, Petr Kůrka, Finite State Transducers of Modular Möbius Number Systems, Branislav Rovan, Vladimiro Sassone, Peter Widmayer (editors), Mathematical Foundations of Computer Science 2012, 37th International Symposium, MFCS 2012, Proceedings, Springer, LNCS 7464, page 323,
Modular Möbius number systems consist of Möbius transformations with integer coefficients and unit determinant.

Classical Kleinian groups are discrete subgroups of Möbius transformations which act on the Riemann sphere with a nonempty region of discontinuity.

2013, Angel Cano, Juan Pablo Navarrete, Seade Kuri José Antonio, Complex Kleinian Groups, page 1:

complex projective line
complex projective line

(geometry, complex analysis) A transformation of the extended complex plane that is a rational function of the form f(z) = (az + b) / (cz + d), where a, b, c, d are complex numbers such that ad − bc ≠ 0; an automorphism of the complex projective line.

August Ferdinand Möbius
Möbius transformation


point at infinity

point at infinity (plural points at infinity)

1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45,
In the geometry of inversion, therefore, it is usual to sacrifice the line of points at infinity which is so useful in other fields, and adopt the convention of a single point at infinity, the inverse of the center of the circle of inversion.

As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.

1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10:

1997, Susan Addington, Stuart Levy, Lost in the Fun House: An Application of Dynamic Projective Geometry, James King, Doris Schattschneider, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, page 159,
To unify the treatment of concurrent and parallel lines, define a point at infinity for each family of parallel lines, and declare each point at infinity to lie on each of the parallel lines that define it. […] Now parallel lines meet at infinity, and vanishing points are the images of points at infinity.

Projective space is, by definition, the collection of the points in affine space together with the points at infinity.

2003, Ron Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, page 18:

(geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion.

Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.

1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:

2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,
As we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.
This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré, […] .

To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.

2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:

(geometry, hyperbolic geometry) An ideal point.


circle inversion

circle inversion (plural circle inversions)

The first of these is obtained by the method of circle inversion first discovered by Peaucellier.

1875, A. B. Kempe, On Some New Limkages: Messenger of Mathematics, volume 4, page 121:

Nikolai Joukowski (1847-1921) was a Russian mathematician who did research in aerodynamics (James and James, 1992). He used circle inversion in the complex number plane to study airfoil shapes.

1997, Mathematical Association of America, MAA Notes, numbers 41-42, page 169:

As is common with nonlinear projections, it is possible to come up with many variants of circle inversions.[…]An obvious (but perhaps not very useful) extension of circle inversion is sphere inversion, where the spaces inside and outside a sphere are swapped.

2011, David Salomon, The Computer Graphics Manual, page 88:

(geometry, inversive geometry, for a given circle with centre O and radius r) A transformation that maps any point P ≠ O to the point P’ on the ray from O through P for which OP × OP’ = r², and which maps O to the point at infinity.


Morley triangle

Morley triangle (plural Morley triangles)

Named for Anglo-American mathematician Frank Morley.

Show that the sides of the Morley triangle of T (cf. 10.3.10) lie on three of those lines (notice that each vertex of the Morley triangle is the center of a cardioid lying inside T and tangent to all its sides, one of them twice.)

2009, Marcel Berger, Geometry I, page 276:

The length of a side of the Morley triangle of a given triangle T is less than one third the length of the smallest side of T.
Remark. The Morley triangle of T is the equilateral triangle formed by the intersection in pairs of the angle trisectors of T.

2013, Dragoslav S. Mitrinovic, J. Pecaric, V. Volenec, Recent Advances in Geometric Inequalities, page 276:

There are a total of 18 Morley triangles that can be constructed. One amusing article I spotted on the web was from a math enthusiast who wrote a computer program trying to illustrate the central Morley triangle we started with, but due to a bug actually trisected the exterior angles in come cases... and was surprised to produce equilateral Morley triangles anyway!

2016, Erik Seligman, Math Mutation Classics, page 27:

Morley's trisector theorem
Morley's trisector theorem

equilateral triangle
equilateral triangle

(geometry) Any of a number of triangles described in Morley's trisector theorem, especially the equilateral triangle formed from the points of intersection of adjacent angle trisectors of a given triangle.

triangle de Morley

first Morley triangle, Morley's triangle

equilateral triangle formed by intersection points of adjacent trisectors


Exeter point

Named for the Phillips Exeter Academy (in the town of Exeter, New Hampshire), where it was discovered during a computers-in-mathematics workshop in 1986.

It ^([the Euler line]) passes through the centroid, the Exeter point and the center of the nine-point circle of the triangle.

2015, Gerard Venema, Just the Facts 101: Foundations of Geometry, Cram101 Textbook Reviews, unnumbered page:

(geometry, given a triangle ΔABC whose medians intersect its circumcircle at {A, A’}, {B, B’} and {C, C’} and ΔDEF constructed from the tangent lines at A, B and C such that D, E and F are respectively opposite A, B and C) The point of concurrence of the rays DA’, EB’ and FC’.

B and C such that D; E and F are respectively opposite A; B and C; B and C such that D; E and F are respectively opposite A; B and C

Clark Kimberling
Encyclopedia of Triangle Centers
Exeter point
Exeter, New Hampshire
Phillips Exeter Academy


Euler line

Named for Swiss mathematician Leonhard Euler.

Since the line joining the circumcenter and orthocenter of a triangle is its Euler line, we see that this parabola is the envelope of the Euler lines of the triangles Aᵢ.

1983, The American Mathematical Monthly, volume 40, page 199:

The Euler line in the preceding figure is OH. N, the center of the nine-point circle, not only lies on the Euler line, but is also its midpoint.

2000, Alfred S. Posamentier, Making Geometry Come Alive: Student Activities and Teacher Notes, page 147:

2011, Derek Allan Holton, A Second Step to Mathematical Olympiad Problems, World Scientific Publishing, Mathematical Olympiad Series, Volume 7, page 57,
Show that the perpendicular bisector of LM in Figure 2.5 meets the Euler line halfway between the orthocentre and the circumcentre of ΔABC.

(geometry) A line that, for a given triangle, passes through several important points, including the circumcentre, orthocentre and centroid; an analogous line for certain other 2- and 3-dimensional geometric figures.


right trapezoid

right trapezoid (plural right trapezoids)

Right trapezoids are used in the trapezoidal rule for estimating areas under a curve.

(geometry) A trapezoid with two adjacent right angles


inversive geometry

inversive geometry (countable and uncountable, plural inversive geometries)

1983, John Willard Milnor, On the Geometry of the Kepler Problem, The American Mathematical Monthly, Volume 90, reprinted in 1994, John Milnor, Collected Papers, I: Geometry, page 261,
In particular, the non-Euclidean geometry of Lobachevsky and Bolyai, and the inversive geometry of Steiner play an important role.

The study of properties of such families of circles gave rise to a new geometry, called inversive geometry, which was able to provide particularly striking proofs of previously known results in Euclidean geometry as well as new results.

1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 2:

Even after that, despite the need identified by Klein, Kasner, and J. W. Young, foundations of inversive geometry did not receive much research attention.
David Hilbert's school at Göttingen spawned an axiomatization of inversive geometry by B. L. van der Waerden and Lucas J. Smid in 1935.

2007, Elena Anne Marchisotto, James T. Smith, The Legacy of Mario Pieri in Geometry and Arithmetic, page 143:

(geometry) The branch of geometry concerned with inversion transformations, specifically circle inversions in the Euclidean plane, but also as generalised in non-Euclidean and higher-dimensional spaces.


midcircle

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Every triangle has a midcircle, and its center is the fourth kind of middle point that a triangle can have. In 1767 Leonhard Euler proved that for all triangles, the orthocenter, the circumcenter, the centroid and the center of the midcircle are always on the same line.

2010, Alex Bellos, Here's Looking at Euclid, page 61:

(geometry) The circle (if one exists) that passes through the midpoint of each side of a given polygon, especially a triangle.


semimean

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The planes in which lie the semimajor and semimean axes of both the pebbles and quartz-feldspar aggregates coincide with those of the crystalline schistosity (Sj, which follows the bedding (S0).

1970, Akademii͡a︡ nauk SSSR, Doklady of the Academy of Sciences of the U.S.S.R:

(geometry) Pertaining to half the length of the middle axis on an ellipse (halfway between the major and minor axes).


chainette

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hexlet

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(geometry) A chain of six spheres, each of which is tangent to both of its neighbours and also to three mutually tangent given spheres.


subcomb

A corresponding series of descending subcombs lay in place, each successive subcomb's conduits, or teeth, having diameters decreased by half.

2007, Ron Sanders, Microcosmia, →ISBN, page 132:

With these choices, a subcomb with at least one tooth can be smoothed.

2013, Olivier Debarre, Higher-Dimensional Algebraic Geometry, →ISBN, page 106:

(algebraic geometry) A subcurve of a comb that includes the handle.


cubinder

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(geometry) A cubical cylinder (four-dimensional analog of a cylinder)

(geometry) An irregular prism that is square at one end but circular at the other


horodisk

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In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space #92;mathbb#123;H#125;ⁿ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain #92;Omega#92;subset#92;mathbb#123;H#125;² with C² boundary whose complement is connected and supports a bounded positive solution u to an overdetermined problem, assuming natural conditions on the equation and the behaviour at infinity of the solution, must be either a geodesic ball or, a horodisk or, a half-space determined by a complete equidistant curve or, the complement of any of the above example.

2015, José M. Espinar, Alberto Farina, Laurent Mazet, “f-extremal domains in hyperbolic space”, in arXiv:


hyperradius

hyperradius (plural hyperradii or hyperradiuses)

hyperradii (plural)
hyperradiuses (plural)

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subface

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At each iteration, our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum-norm subgradient of the objective function or execute a CG subroutine that conducts a sequence of CG iterations until a CG iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found.

2015, Zhaosong Lu, Xiaojun Chen, “Generalized Conjugate Gradient Methods for ℓ₁ Regularized Convex Quadratic Programming with Finite Convergence”, in arXiv:


semiampleness

(algebraic geometry) The quality of being semiample.


semiample

(algebraic geometry) Having a positive multiple that is generated by global sections.


nefness

We also get some general results under nefness and relative semi-ampleness assumptions..

2016, Lei Zhang, “Subadditivity of Kodaira dimensions for fibrations from minimal models to projective varieties of general type”, in arXiv:

(algebraic geometry) The quality of being nef.


semiarc

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hypercuboid

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(geometry) An analog of a cuboid in four or more dimensions

nadkvádr

analogue of a cuboid in four or more dimensions


paracomplex

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Applications include complex, paracomplex and symplectic geometries..

2016, Giovanni Calvaruso, Gabriela P. Ovando, “From almost (para)-complex structures to affine structures on Lie groups”, in arXiv:


equiregular

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(geometry) Having both all sides equal length and all angles equal.


plesiohedron

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The plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron.

(geometry) A special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps.


rep-tile

Coined as a pun on reptile by US mathematician Solomon W. Golomb. See repeat and tile.

(geometry) A shape that can be dissected into a number of smaller copies of the same shape.


octacube

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symptosis

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(geometry) The meeting of polars of the same point with reference to different loci.


centerpoint

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(statistics, geometry) A generalization of the median to data in higher-dimensional Euclidean space.


Cremona-Richmond configuration

It was studied by Cremona (1877) and Richmond (1900).

(geometry) A configuration of 15 lines and 15 points, having three points on each line and three lines through each point, and containing no triangles.


Heegner point

Named after Kurt Heegner, German mathematician.

(geometry) A point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane.


taxidistance

taxidistance (countable and uncountable, plural taxidistances)

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Taxidistance is greater than Euclidean Distance.

1970 March, F Papy, G Papy, “Taximetry”, in International Journal of Mathematical Education in Science, volume 1, number 4:

Although she was still ready to support her previous result — that her diamond represented all the points at the same taxidistance from the center point — she also began to realize that this figure did not satisfy many of the other properties of circles.

1992, Raffaella Borasi, Learning mathematics through inquiry, page 86:

Help the students understand that there are multiple pathways between two points which have the minimum taxidistance.

1997, HRW geometry: explore, communicate, apply. Teaching transparencies directory, →ISBN:

taxicab distance
taxicab distance#English

(geometry) Alternative form of taxicab distance


taxi distance

taxi distance (countable and uncountable, plural taxi distances)

What is the taxi distance from A to P? Mark all the points you can find at a taxi distance 4 from A. Connect these points to form a square. This figure is a taxi circle.

2000, Alfred S. Posamentier, Making Geometry Come Alive: Student Activities and Teacher Notes, →ISBN:

To find the taxi distance between your home and your school, use the Ruler Postulate to find the horizontal and vertical distances.

2007, Orlando A. Oronce, Marilyn O. Mendoza, E-math: Geometry, →ISBN, page 423:

The taxi distance from P to Q is 7.

2012, EF Krause, Taxicab geometry: An adventure in non-Euclidean geometry:


superdisk

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(geometry) A closed curve, of form y = (1-xⁿ)^(1/n) (where n > 2), having properties intermediate between those of a circle and a square


halfsphere

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irregular prism

irregular prism (plural irregular prisms)

A right prism, having regular polygons for its terminating planes, is said to be regular ; an irregular prism is one in which the ends are irregular polygons.

1867, William Chambers, Robert Chambers, Chambers's Information for the People, page 652:

If the base be an irregular polygon of any number of sides, the prism is then called an irregular prism.

1806, Peter Jonas, The Genuine Art of Gauging Made Easy and Familiar, page 138:

(geometry) a prism whose cross section is an irregular polygon

If the base be a triangle, it is called a triangular prism; if a square, a square prism; if a pentagon, a pentagonal prism; and' if the base is any irregular figure, it is called an irregular prism : ...

1850, John Radford Young, An introductory treatise on mensuration, in theory and practice, page 115:

Irregular prisms have a cross-section with no well known shape. Some of the steel sections used in the construction industry belong to this category.

2016, Kusum Wadhwa, Anju Loomba, Saraswati Mathematics, Saraswati House Pvt Ltd, →ISBN, page 234:

(geometry) a solid that is like a prism but whose cross section is not (necessarily) a polygon


circle of antisimilitude

circle of antisimilitude (plural circles of antisimilitude)

(geometry) A reference circle about which two other circles are inverses of each other


cubohemioctahedron

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(geometry) A faceted form of the cuboctahedron.

立方半八面体

りっぽうはんはちめんたい


exinscribed

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(geometry) inscribed inside a polygon (and touching every side of it)


golyhedron

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(geometry) A three-dimensional generalization of a golygon.


mirror symmetry

(algebraic geometry, theoretical physics) A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.


rectanguloid

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(geometry) A rectangular object; A three-dimensional form related to the rectangle


spheroform

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(geometry) A three-dimensional, convex form that has a constant thickness


pentatope

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nonhypotenuse

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(geometry) Either of the two smaller sides of a right-angled triangle


unode

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(geometry) A double point of a surface whose tangent cone consists of one double plane.


logarithmic spiral

logarithmic spiral (plural logarithmic spirals)

(geometry) A self-similar spiral curve which often appears in nature.


golden spiral

(geometry) A logarithmic spiral whose growth factor is φ, the golden ratio.


position vector

position vector (plural position vectors)

Coordinate term: direction vector

(geometry) Of a point, a vector going from the origin to that point; equivalently, the transpose of the point.


quadricone

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Gauss map

Named after German mathematician Carl Friedrich Gauss.

1969 [Van Nostrand], Robert Osserman, A Survey of Minimal Surfaces, 2014, Dover, Unabridged republication, page 73,
There exist complete generalized minimal surfaces, not lying in a plane, whose Gauss map lies in an arbitrarily small neighborhood on the sphere.

1985, R. G. Burns (translator), B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry― Methods and Applications: Part II: The Geometry and Topology of Manifolds, Springer, Graduate Texts in Mathematics, page 114,
14.2.2 Theorem The integral of the Gaussian curvature over a closed hypersurface in Euclidean n-space is equal to the degree of the Gauss map of the surface, multiplied by γₙ (the Euclidean volume of the unit (n-1)-sphere).

In §2 we use results of §1 for the study of Gauss maps of projective varieties. The classical Gauss map associates to each point of a nonsingular real affine hypersurface the unit vector of the external normal at this point.

2005, F. L. Zak, Tangents and Secants of Algebraic Varieties, American Mathematical Society, page 5:

differential geometry
differential geometry

(geometry, differential geometry) A map from a given oriented surface in Euclidean space to the unit sphere which maps each point on the surface to a unit vector orthogonal to the surface at that point.


turning number

This program illustrates the Whitney-Graustein theorem, that two closed curves (immersions of the circle) in the plane can be deformed into one another without letting go of the immersion property if and only if their turning numbers are the same.

1989, Wayne Sewell, Weaving a Program: Literate Programming in WEB, page 516:

Moreover, l gives the turning number of the k-gon one sees in the horizontal symmetry plane when the axes of the ends are deleted.[…]Moreover, every even k ≥ 30 except for 32, 36, 40, 44, 48 52 an 56 gives an example, while for every even k > 102 there is more than one example since different turning numbers are possible.

1998, Karsten Groβe-Brauckmann, Robert B. Kusner, John M. Sullivan, “Constant Mean Curvature Surfaces with Cylindrical Ends”, in Hans-Christian Hege, Konrad Polthier, editors, Mathematical Visualization: Algorithms, Applications and Numerics, Springer, page 114:

2002, Journal of Physics: Mathematical and general, page 6187,
For any oriented curve β(τ),τ∈I,I=[0,Λ] of class C¹ (τ is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number 𝒯_(n_β) may be defined as
𝒯_(n_β)=1/(2π)∫₀^Λκ_β(τ)dτ
where κ_β(τ) is the signed curvature of β.

mathematical analysis
mathematical analysis

(geometry, topology, mathematical analysis) A version of winding number in which the number of rotations is counted with respect to the tangent of the curve rather than a fixed point.


spherical cone

(geometry) A cone with a spherical base.


transflection

transflection (countable and uncountable, plural transflections)

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Clearly, A", B", C" are not images of A, B, C, under either a translation, rotation, or reflection. Hopefully, it will be possible to find a unique transflection which does the job.

1968, Myron Frederick Rosskopf, Joan L. Levine, Bruce Ramon Vogeli, Geometry; a Perspective View, page 236:

During the sequence of considering translations, rotations, reflections, and transflections of a plane α, we show that ...

1972, Merlyn J. Behr, Dale G. Jungst, Fundamentals of elementary mathematics; geometry, volume 2, page 261:

Visualize these operations by considering the simple screw and transflection. Transflection can be imagined as seeing yourself in a mirror even as the mirror is being moved parallel to itself.

2015, C. S. Sunandana, Introduction to Solid State Ionics: Phenomenology and Applications, →ISBN:

(geometry) A transfiguration consisting of reflection about a line combined with translation along the same line.


trapezian

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(geometry) Having opposed trapeziform faces.


symmedian

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(geometry) Any of three lines associated with every triangle, constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side) and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half).


line of apsides

line of apsides (plural lines of apsides)

(geometry, astronomy) Synonym of apse line


rhombiform

en-adj

rhombiform (comparative more rhombiform, superlative most rhombiform)

more rhombiform (comparative)
most rhombiform (superlative)

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cevian

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(geometry) A line from the vertex of a triangle to a point on the opposite side


isotomic

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From iso- (“same”) + -tome (“cutting, section”) + -ic (“adjectival suffix”), ultimately from Ancient Greek τέμνω (témnō, “I cut”).

affix

(geometry) Having equal but reflected cevians.


enneadecagon

(geometry) A polygon with nineteen sides and nineteen angles.


hexadecagon

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(geometry) A polygon with sixteen sides and sixteen angles.


anisotomic

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icositetragon

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(geometry) A polygon with 24 sides and 24 angles.


hexacontagon

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From Ancient Greek ἑξήκοντα (hexḗkonta, “60”) + -gon, from Ancient Greek γωνία (gōnía, “corner”).


opposite angle

(geometry) Either of the angles directly opposite to each other when two lines intersect in a plane.


adjacent angle

(geometry) Either of a pair of angles that share a common side


tetradecahedron

en-noun

tetradecahedron (plural tetradecahedra or tetradecahedrons)

tetradecahedra (plural)
tetradecahedrons (plural)

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(geometry) A polyhedron with fourteen faces.


Teichmüller space

Teichmüller space (plural Teichmüller spaces)

(geometry) A space T(S) assigned to a (real) topological (or differential) surface S: it parameterizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism.


Grothendieck-Riemann-Roch theorem

Hirzebruch-Riemann-Roch theorem
Hirzebruch-Riemann-Roch theorem

(algebraic geometry) An important generalization of the Hirzebruch-Riemann-Roch theorem about complex manifolds.


prospectivity

prospectivity (countable and uncountable, plural prospectivities)

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Schur bases his definitions of equality, of addition and multiplication of projective segments, upon the correspondence known as ' prospectivity,' and, at first, avails himself only of the axioms of connection and of order [Shur's postulates 1. to 8.];

1919, Ludwik Silberstein, Projective Vector Algebra:

If a point O on I is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line I; in fact they are all the same prospectivity, however m, n, S, and S' may have been varied subject to these conditions.

2014, Alfred North Whitehead, Science and Philosophy, →ISBN:

Alternatively, Friedrich Schur (1881, p. 253) used Thomae's (1873, p. 11) definition of projectivity in terms of prospectivity to prove the fundamental theorem in a manner which is independent of the Archimedean axiom.

2016, Francesca Biagioli, Space, Number, and Geometry from Helmholtz to Cassirer, →ISBN, page 122:

(geometry) A projective relationship between a line and itself by projecting onto an intersecting line from a point on another line that intersects at the same point, and then back on to the first line from a different point on the first intersecting line.


algebraic variety

algebraic variety (plural algebraic varieties)

If #92;mathcal#123;W#125; is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x#92;in#92;mathcal#123;W#125; axe taken to be the germs of functions on #92;mathcal#123;W#125; induced by functions in the stalk #92;mathfrak#123;R#125;#95;x. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,#92;mathfrak#123;R#125;), by definition, is a local-ringed space.

2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87:

The quest for understanding the structure of algebraic varieties is a very natural one.[…]A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.

2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii:

Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.

2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7:

(algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition.

varisto

set of solutions to a system of polynomial equations; any of certain generalisations of such a set

affine variety

set of solutions to a system of polynomial equations, any of certain generalisations of such a set


algebraic subvariety

algebraic subvariety (plural algebraic subvarieties)

(algebraic geometry) A subset of an algebraic variety that is itself an algebraic variety.


circular angle

(geometry) An angle that defines a position on a circle.


generic property

generic property (plural generic properties)

1989, Thomas S. Parker, Leon O. Chua, Appendix C: Differential Topology, Practical Numerical Algorithms for Chaotic Systems, Springer, Softcover reprint, page 314,
A property P that refers to members of a set Y is a generic property if the subset of Y whose members exhibit property P contains a dense open subset of Y.
Remark: If P₁ and P₂ are generic properties of a set Y, then so is the property "P₁ and P₂."

2001, Constantin I. Chueshov (translator), Igor Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, [1999, Russian edition] Acta Scientific Publishing House, page 117,
It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property.
However, it should be kept in mind that the generic property is not the one that holds almost always. […] Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.

We now formulate the first generic property concerning the singularities of a Hamiltonian function.

2010, H. W. Broer, F. Takens, “Chapter 1: Preliminaries of Dynamical Systems Theory”, in B. Hasselblatt, H. W. Broer, F. Takens, editors, Handbook of Dynamical Systems, volume 3, Elsevier (North-Holland), page 24:

(topology, algebraic geometry) A property that is true in some dense open subset of a given set.


scheme theory

This book develops Grothendieck's scheme theory as a method for studying algebraic geometry.[…]In the preface of EGA [ Éléments de géométrie algébrique ], Grothendieck even claimed that a knowledge of classical algebraic geometry may hinder the reader from studying scheme theory.

1997, Kenji Ueno, translated by Goro Kato, Algebraic Geometry 1: From Algebraic Varieties to Schemes, American Mathematical Society, page x:

There is another approach dealing with more ring theoretical objects, i.e. starting from a nice class of algebras, called schematic algebras, it is possible to define a space, a Zariski topology, points, lines etc..., cf. [VOW1], [AZ], [ATV]. then it becomes possible to develop a scheme theory, completely extending the theory of schemes in classical algebraic geometry.

1999, Freddy Van Oystaeyen, “A Deformation of Projective Schemes”, in Freddy Van Oystaeyen, editor, Commutative Algebra and Algebraic Geometry, Marcel Dekker, page 295:

The main point of scheme theory is to consider C as a covariant functor R#92;mapstoC(R) from the category #92;textit#123;B-ALG#125; (also written as #92;textit#123;ALG#125;#95;#123;#47;#92;mathfrak#123;B

2004, Haruzo Hida, p-Adic Automorphic Forms on Shimura Varieties, Springer, page 68:

locally ringed space
locally ringed space

(algebraic geometry) The branch of mathematics that concerns schemes (locally ringed spaces admitting coverings by open sets, each isomorphic to the spectrum of some commutative ring).


projective variety

projective variety (plural projective varieties)

Varieties are sometimes called closed sets and some authors call an open subset of a projective variety a quasiprojective variety. The latter term is in an attempt to unify the concept of affine and projective variety.

2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724:

A closed subset V#92;subset#92;mathbb#123;C#125;Pⁿ is called a (complex) projective variety if, locally, V is defined by a set of complex polynomial equations. Outside of its singular locus, that is, away from the subset where the defining equations do not have maximal rank, the projective variety is a complex submanifold of #92;mathbb#123;C#125;Pⁿ.

2006, Werner Ballmann, Lectures on Kähler Manifolds, European Mathematical Society, page 16:

The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic.

2015, Katsutoshi Yamanoi, “Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory”, in Takushiro Ochiai, Toshiki Mabuchi, Yoshiaki Maeda, Junjiro Noguchi, Alan Weinstein, editors, Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi, Springer (Birkhäuser), page 209:

homogeneous polynomial
homogeneous polynomial

(algebraic geometry) A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.


unit hyperbola

(geometry) The hyperbola that satisfies the equation: x²-y²=1


golden triangle

golden triangle (plural golden triangles)

(geometry) An isosceles triangle whose duplicated side is in the golden ratio to the distinct side.


degenerate conic

degenerate conic (plural degenerate conics)

(geometry) A conic section that is reducible as the union of two lines.


group functor

In order to be able to efficiently state the main results of this chapter (in §4) we formulate here a group-functor version of the R-nilpotent tower lemma (Ch.III, 6.4).

1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106:

A direct product of k–group functors is again a k–group functor; so is a fibre product if the morphisms used in its construction are homomorphisms of k–group functors.

2003, Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, American Mathematical Society, page 20:

By a group functor we mean a functor from small k-algebras to groups. A homomorphism of group functors is a natural transformation. A subgroup functor of a group functor G is a subfunctor H such that H(R) is a subgroup of G(R) for all R; it is normal if H(R) is normal in G(R) for all R.

2017, J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, page 118:

(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.


hyperbolic angle

hyperbolic angle (plural hyperbolic angles)

(geometry) An angle that describes a position along a hyperbola.


hyperbolic sector

hyperbolic sector (plural hyperbolic sectors)

rectangular hyperbola
rectangular hyperbola

(geometry) An area on a plane bounded by two rays from the origin and a rectangular hyperbola.


Fibonacci spiral

Fibonacci spiral (plural Fibonacci spirals)

(geometry) An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in a Fibonacci tiling; a spiral formed by connecting quarter-circles whose radii are Fibonacci number multiples of the initial radius.


linear algebraic group

linear algebraic group (plural linear algebraic groups)

Geometric invariant theory arises in an attempt to construct a quotient of an algebraic variety X by an algebraic action of a linear algebraic group G.

2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii:

The differential Galois group of a homogeneous linear differential equation has a structure of linear algebraic group; hence it is endowed, in particular, with the Zariski topology.[…]Kolchin used the differential algebra developed by Ritt and also built the foundations of the theory of linear algebraic groups.

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page xi:

2015, Willem A. de Graaf, Orbit Closures of Linear Algebraic Groups, Jaime Gutierrez, Josef Schicho, Martin Weimann (editors), Computer Algebra and Polynomials: Applications of Algebra and Number Theory, Springer, LNCS: 8942, page 76,
Actions of linear algebraic groups appear in many contexts. […] Throughout we assume that the base field is algebraically closed and of characteristic 0, as many constructions that we use (e.g., the correspondence between a linear algebraic group and its Lie algebra) only work well in characteristic 0.

general linear group
general linear group

(algebraic geometry, category theory) An algebraic group that is isomorphic to a subgroup of some general linear group.


algebraic group

algebraic group (plural algebraic groups)

The purpose of this article is to discuss various results concerning the subgroups of simple algebraic groups G and of the corresponding finite groups of Lie type G^F (where F is a Frobenius morphism).

1998, Martin W. Liebeck, “Introduction to the subgroup structure of algebraic groups”, in Roger William Carter, Meinolf Geck, H. K. Moffatt, editors, Representations of Reductive Groups, Cambridge University Press, page 129:

We begin by showing how to make the quotient of an affine algebraic group by a normal algebraic subgroup into an affine algebraic group, and we determine the structure of commutative affine algebraic groups.

1998, Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, page 464:

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page 75,
We point out that some properties of an algebraic group are more easily read in its Lie algebra. An important property of an algebraic group is its solvability.

(algebraic geometry) A group that is an algebraic variety.


regular map

Morphism of algebraic varieties
Regular map (graph theory)

We now find out how a regular map acts on the ring of regular functions on a closed set.[…]Moreover, regular maps can be characterised as the maps that take regular functions into regular functions.

1994, Igor R. Shafarevich, translated by Miles Reid, Basic Algebraic Geometry 1, 2nd edition, Springer, page 29:

Since one expects that the canonical projection f#58;X#92;rightarrowX#47;G is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of X.

2017, José F. Fernando, José M. Gamboa, Carlos Ueno, Polynomial, regular and Nash images of Euclidean spaces, Fabrizio Broglia, Françoise Delon, Max Dickmann, Danielle Gondard-Cozette, Victoria Ann Powers (editors), Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 160,
The 1-dimensional semialgebraic set 𝒮:=x>0,xy=1 is the image of the regular map
f: R ²→ R ²,(x,y)↦((xy-1)²+x²,1/((xy-1)²+x²)).

(algebraic geometry) A morphism between algebraic varieties.


nonacoptic

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From non- + acoptic, introduced by the mathematician Branko Grünbaum to clarify terminology in the geometry of polygons and polyhedra. Initially he used the term coptic, but abandoned it in favor of nonacoptic in later writings.

1999, Grünbaum, Branko, "Acoptic polyhedra", Advances in discrete and computational geometry (South Hadley, MA, 1996) (PDF), Contemp. Math., 223, Amer. Math. Soc., Providence, RI, pp. 163–199, doi:10.1090/conm/223/03137, MR 1661382.
See Norgate [N2], where nonacoptic pentahedra are investigated as well.

Branko Grünbaum
small stellated dodecahedron


acoptic

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Introduced by the mathematician Branko Grünbaum to clarify terminology in the geometry of polygons and polyhedra. From Ancient Greek ἀ- (a-, “not”) + κόπτω (kóptō, “to cut”).

affix

1999, Branko Grünbaum, "Acoptic polyhedra", Advances in discrete and computational geometry (South Hadley, MA, 1996) (PDF), Contemp. Math., 223, Amer. Math. Soc., Providence, RI, pp. 163–199, doi:10.1090/conm/223/03137, MR 1661382.

(geometry, uncommon) Not self-intersecting.


semiparabolic

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(geometry) Having the form of a semiparabola.


cocircular

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(geometry, of a set of points) Lying on the same circle


cocircularity

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(geometry) The condition of being cocircular


field extension

en-noun

field extension (plural field extensions)

A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).

1974, Thomas W. Hungerford, Algebra, Springer, page 230:

1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of ψ includes at most a finite constant field extension.

2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.

kuntalaajennus

pair of fields such that one is a subfield of the other


Plücker coordinate

Plücker coordinate (plural Plücker coordinates)

Introduced by Julius Plücker in the 19th century.

(geometry) Any of six homogeneous coordinates assigned to a line in projective 3-space, used in geometric algebra and computer graphics.


Malfatti circle

Malfatti circle (plural Malfatti circles)

Named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.

(geometry) Any of the three circles inside a given triangle such that each circle is tangent to the others and to two sides of the triangle.


vertical angle

vertex angle

(geometry) Any of the opposite angles formed by intersecting lines.

vrcholový úhel

(geometry) The angle at the apex of an isosceles triangle or cone.


tetrahelix

tetrahelix (plural tetrahelixes or tetrahelices)

tetrahelixes (plural)
tetrahelices (plural)

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(geometry) A helix formed from tetrahedra linked face to face.


finite geometry

finite geometry (countable and uncountable, plural finite geometries)

But the work of von Staudt, Fano, and Veblen led even to the construction of finite geometries having only finitely many points and lines. Originally having only incidence relations, these finite geometries have been extensively developed through the introduction of suitable axioms of order and congruence by Järnefelt and Kustaanheimo, who have proposed their use in physics to solve certain paradoxes due apparently to the use of continuous variables.

1980, Judson Chambers Webb, Mechanism, Mentalism and Metamathematics: An Essay on Finitism, Springer, page 39:

A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed.

1999, S. Ball, “Polynomials in Finite Geometries”, in J. D. Lamb, D. A. Preece, editors, Surveys in Combinatorics, 1999, Cambridge University Press, page 17:

Finite geometries, such as Euclidean and projective geometries, are powerful mathematical tools for constructing error-control codes.

2009, William Ryan, Shu Lin, Channel Codes: Classical and Modern, Cambridge University Press, page 430:

(geometry) Any geometric system that has only a finite number of points.

The coordinates of geometry were as incompatible with representation theory as were the splitting fields of representation theory with finite geometry.

1983, Robert A. Liebler, “Combinatorial Representation Theory and Translation Planes”, in Norman L. Johnson, Michael J. Kallaher, Calvin T. Long, editors, Finite Geometries: Proceedings of a Conference, CRC Press, page 307:

We give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets.

1996, Dieter Jungnickel, “Maximal Sets of Mutually Orthogonal Latin Squares”, in S. Cohen, H. Niederreiter, editors, Finite Fields and Applications: Proceedings of the 3rd International Conference, Cambridge University Press, page 129:

Finite geometry is a broad area of research, but much of it is not Euclidean in flavour and no theory seems to match that of oriented matroids in establishing combinatorial Euclidean geometry.

2007, Michael B. Smyth, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian E. Pratt-Hartmann, Johan F. A. K. van Benthem, editors, Handbook of Spatial Logics, Springer, page 787:

(geometry, uncountable) The branch of geometry that concerns geometric systems with only a finite number of points.


imaginary geometry

Calque of Russian мни́мая геометрия (mnímaja geometrija), coined by mathematician Nicolai Lobachevsky, probably influenced by мни́мое чи́сло (mnímoje číslo, “imaginary number”), although his precise reasoning is unknown.

calque

мни́мая геометрия

1879, Frederick Pollock, Introduction, Part 1: Biographical, Leslie Stephen, Frederick Pollock (editors), William Kingdon Clifford, Lectures and Essays, Volume 1, Cambridge University Press, page 14,
He liked talking about imaginary geometry, as a matter of pure amusement, to anyone interested in it. But at the same time he attached a serious import to it.

Claim 7. The Euclidean law of cosines holds for any infinitesimal triangle #92;triangleABC in imaginary geometry.

2011, Seth Braver, Lobachevski Illuminated, Mathematical Association of America, page 188:

non-Euclidean geometries
non-Euclidean geometry

(geometry) Absolute geometry, an axiomatised geometry in which the parallel postulate is absent and not replaced by an alternative, and of which Euclidean geometry and some non-Euclidean geometries are subtypes.

мни́мая геометрия

geometry with no parallel postulate — see also absolute geometry


semimeridian

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surface of revolution
surface of revolution

(geometry) The intersection of a surface of revolution and a half-plane through its centre


complex projective line

en-noun

complex projective line (plural complex projective lines)

By way of an example, we consider in detail the complex projective line #92;C#92;mathbb#123;P#125;¹.[…]Thus via the function w#95;0 the complex projective line #92;C#92;mathbb#123;P#125;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point at infinity").
2..2.1 Theorem The complex projective line #92;C#92;mathbb#123;P#125;¹ is diffeomorphic to the 2-dimensional sphere S².

1985, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, translated by R. G. Burns, Modern Geometry — Methods and Applications: Part II, Springer, page 18:

Finally, C is interested in the geometry of the complex projective line P¹(#92;C).

2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:

Hence, C(#92;Gamma) is an uncountable union of complex projective lines.

2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142:

As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #92;dot#123;S#125;² ̇provides a conformal model for the complex projective line #92;C#92;mathbb#123;P#125;¹.

2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194:

equivalence relation
equivalence relation

(analytic geometry, projective geometry) A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation "(α, β) ≡ (λα, λβ) for all nonzero complex λ".


complex line

1990, R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Volume 1 Function Theory, Wadsworth & Brooks/Cole, page 102,
However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in C ⁿ through a point A∈ C ⁿ in the direction of a vector B∈ C ⁿ is the one-dimensional complex submanifold of C ⁿ described parametrically as A+tB:t∈ C.

A complex line in #92;Cⁿ is a set of the form l#61;#92;#123;z#58;z#61;a#43;#92;lambdab,#92;lambda#92;in#92;C#92;#125;, where a and b are fixed points in #92;Cⁿ, with b#92;ne 0. Let us say that l is the complex line through a in the “direction” b.

2014, Paul M. Gauthier, Lectures on Several Complex Variables, Springer (Birkhäuser), page 39:

The question of finding different familes^([sic]) of complex lines sufficient for holomorphic extension was put in [12]. Clearly, the family of complex lines passing through one point is not enough. As shown in [16], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient.

2018, Bairambay Otemuratov, “A Mulitidimensional Analogue of Hartogs's Theorem on n-Circular Domains for Integrable Functions”, in Zair Ibragimov, Norman Levenberg, Utkir Rozikov, Azimbay Sadullaev, editors, Algebra, Complex Analysis, and Pluripotential Theory: 2 USUZCAMP, 2017, Springer,, page 110:

(complex analysis, analytic geometry) A 1-dimensional affine subspace of a vector space over the complex numbers.

complex plane

affine subspace that is 1-dimensional over the complex numbers


trilobular

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(geometry) Having a shape that is broadly triangular but is rounded, with three lobular segments conjoined.


quadrisector

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(geometry) A line or curve that quadrisects something.


equidissection

equidissection (countable and uncountable, plural equidissections)

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(geometry) The division of a polygon into triangles of equal area.


Monsky's theorem

(geometry) A theorem stating that it is not possible to dissect a square into an odd number of triangles of equal area.


equidissectable

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(geometry) Of a polygon: able to be divided into triangles of equal area.


spiric

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(geometry) A curve, the plane section of a torus.


shape operator

Gaussian curvature is a prime example; although defined in terms of shape operators, it belongs to this intrinsic geometry, since it passes the test of isometric invariance.

1966, Barrett O'Neill, Elementary Differential Geometry, Academic Press, page 304:

Timelike surfaces have symmetric shape operators which can be put into one of three canonical forms on a fixed tangent space with respect to an orthonormal basis:[…].

2006, Balkan Journal of Geometry and Its Applications, volumes 11-12, page 41:

We have already seen that a plane has zero shape operator. Intuitively, since the shape operator detects the change in the unit normal U, a zero shape operator for a surface M should imply that M is a plane. This is verified by the following result.

2007, John Oprea, Differential Geometry and Its Applications, Mathematical Association of America, page 85:

(geometry, differential geometry) The differential of the Gauss map of an oriented surface at a given point on the surface.


conjugate axis

(geometry) Either of two axes in a conic section, such that each is parallel to the tangent at the extremity of the other.


geometric analysis

Papers are submitted for publication by sending three hard (printed) copies to The Journal of Geometric Analysis, Department of Mathematics, Washington University,[…].

1999, The Journal of Geometric Analysis, volume 9, CRC Press, page 273:

2013, Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, Gilbert Weindtein (editors) Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, American Mathematical Society.

In 2012, thematic programs on Geometric Analysis and Spectral Theory and on Moduli spaces, Extremality and Global Invariants were held at the CRM. There were several workshops on cutting-edge topics in geometric analysis and spectral theory, notably on the Geometry of Eigenvalues and Eigenfunctions, on Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis and on Spectral invariants on non-compact and singular spaces.

2014, Pierre Albin, Dmitry Jakobson, Frédéric Rochon, “Preface”, in Geometric and Spectral Analysis, American Mathematical Society, page vii:

partial differential equation
partial differential equation

(mathematical analysis, geometry, topology) A branch of mathematics in which techniques from differential geometry are used in the analysis and solution of partial differential equations, and vice versa.


witch of Agnesi

witch of Agnesi (plural witches of Agnesi)

Named after Italian mathematician Maria Gaetana Agnesi, who wrote about such curves, and from a mistranslation of Italian versiera, the term she used for them which means "rope", but which is spelled the same as versiera (“witch”).

(geometry) A cubic plane curve defined from two diametrically opposite points of a circle.


angle of torsion

angle of torsion (plural angles of torsion)

(geometry) The indefinitely small angle between two consecutive osculating planes of a curve of double curvature.


semiordinate

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(geometry) Half a chord bisected by the transverse diameter of a conic.


Hurwitz surface

Hurwitz surface (plural Hurwitz surfaces)

(geometry) A compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms.


Chern class

(mathematics, topology, geometry) Any of a group of characteristic classes associated with complex vector bundles, with applications in physics, string theory, etc.


semiconcentric

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(geometry) Having the form of concentric semicircles


projective space

projective space (plural projective spaces)

(geometry) A kind of geometrical space in which only relation(s) of incidence between points and lines matter; in such a space distances or angles do not matter and parallelism (as in an affine space) does not matter either.


affinely independent

linearly independent
linearly independent

(geometry) Said of a set of points in an affine space: the property that the vectors issuing from an arbitrarily chosen point to the rest of the points are linearly independent.


projective frame

projective frame (plural projective frames)

affinely independent
affinely independent

(geometry) The projection onto an n − 1 dimensional hyperplane (which is extended so as to be projective) of a set of n + 1 affinely independent n-dimensional points (called an affine frame).


real projective plane

real projective plane (plural real projective planes)

(geometry) A three-dimensional vector space over the field of real numbers in which one-dimensional linear subspaces are considered as “points” and two-dimensional linear subspaces are considered as “lines”. Equivalently, a sphere in which pairs of antipodal points are considered as “points” and great circles are considered as “lines”.


homogeneous coordinate

homogeneous coordinate (plural homogeneous coordinates)

(geometry) Any one of a set of n + 1 coordinates which non-uniquely identify a point in an n-dimensional projective space; when such coordinates are all multiplied by the same scalar, then the set of them still identifies the same point.


rhombidodecahedral

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(geometry) Relating to a rhombidodecahedron


Miquel's theorem

(geometry) A result in geometry concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides.


Kepler-Poinsot polyhedron

Kepler-Poinsot polyhedron (plural Kepler-Poinsot polyhedra)

Named after Johannes Kepler and Louis Poinsot; Poinsot rediscovered Kepler's figures in 1809.


unduloid

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(geometry) A surface of revolution of an elliptic catenary


orthotomic circle

orthotomic circle (plural orthotomic circles)

(geometry) A circle that cuts three other given circles at right angles.


great-circle arc

great-circle arc (plural great-circle arcs)

This causes a greater deviation of the ground trace from a great-circle arc due to the increased effect of Earth's rotation.

2006, Irfan Ali, Pierino G. Bonanni, Naofal Al-Dhahir, John E. Hershey, Doppler Applications in LEO Satellite Communication Systems, Springer Science & Business Media, →ISBN, page 66:

The two axes connected by the solid but incomplete great-circle arc were measured along a single sahlfold axial surface on opposite sides of a tubular fold ; ...

2012, E. Hansen, Strain Facies, Springer Science & Business Media, →ISBN, page 85:

... it may be swung in a meridian-plane, and a great-circle arc having a radial pivot in said meridian-plane, ...

1898, Official Gazette of the United States Patent Office:

Any great-circle arc of length #92;theta between zero and #92;pi is uniquely identified by its two end points, say a and b.

2002, Leo Dorst, Chris Doran, Joan Lasenby, Applications of Geometric Algebra in Computer Science and Engineering, Springer Science & Business Media, →ISBN, page 81:


duoprism

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(geometry) A polytope resulting from the Cartesian product of two polytopes.


pseudorhombic

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(geometry, crystallography) describing a structure that is almost rhombohedral


Radon's theorem

(geometry) A theorem on convex sets, stating that any set of d + 2 points in Rᵈ can be partitioned into two sets whose convex hulls intersect.


Radon point

(geometry) A point in the intersection of convex hulls specified by Radon's theorem.


Helly's theorem

Named after Eduard Helly, who made the discovery in 1913.

(geometry) A basic result in discrete geometry on the intersection of convex sets, which gave rise to the notion of a Helly family.


vanishing ideal

vanishing ideal (plural vanishing ideals)

(algebraic geometry) An ideal consisting of all polynomials (belonging to a certain polynomial ring) which zero out within a given algebraic variety.


intersector

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Delaunay triangulation

Delaunay triangulation (countable and uncountable, plural Delaunay triangulations)

Named after Boris Delaunay for his work on this topic from 1934.

(geometry, computer graphics) For a given set P of discrete points in a plane: a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P).


rotational symmetry

rotational symmetry (plural rotational symmetries)

(geometry) The property a shape has when it looks the same after some rotation by a partial turn.


tridecahedron

tridecahedron (plural tridecahedra or tridecahedrons)

tridecahedra (plural)
tridecahedrons (plural)

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(geometry) A polyhedron with thirteen faces.


octacontagon

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(geometry) A polygon with 80 sides and 80 angles.


enneacontagon

(geometry) A polygon with 90 sides and 90 angles.


subtendent

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Lastly, if you put BC the lesser of the sides, including the right angle for Radius, the greater side, AC, including the right angle is the tangent of ABC, the greater acute angle opposite; and the subtendent A B, is the secant of the same acute angle.

1642, Bartholomäus Pitiscus, Trigonometry: Or, The Doctrine of Triangles., page 91:

The very Subtendent line is found of an odde part; but the very subtendent of an even part is not found at one operation, but onely the Square of the Subtendent; and the greater the number of parts be, so much harder it willbe to finde out the subtendent.

1657, Johannes Jonstonus, An History of the Constancy of Nature:

The first figure, Vale, hath but one mood, and therefore of as great extent as it selfe, which is Upalem; whose nature is to let us know, whan a plane right angled triangle is given us to resolve, whose subtendent and one of the obliques is proposed, and one of the ambients required, that we must have recourse unto its resolver, which being Rad_U_Sapy→Yr sheweth, that if we joyne the artificiall sine of the angle opposite to the die demanded with the Logarithm of the subtendent, the summe searched in the canon of absolute numbers will afford us the Logarithm of the side required.

1834, Thomas Urquhart, Lord Thomas Maitland Dundrennan, The Works of Sir Thomas Urquhart of Cromarty, Knight, page 81:

For example, even if there were no possible quantity, mathematicians could say that if it was possible to make a right angled triangle, it was something assured that the hypotenuse or the subtendent of the right angle would make a square equal to the squares of the other two sides

1994, Alistair Cameron Crombie, Styles of Scientific Thinking in the European Tradition, page 840:

(geometry, trigonometry) The side that subtends (is opposite) a specified angle;


subtendent

en-adj

subtendent (comparative more subtendent, superlative most subtendent)

more subtendent (comparative)
most subtendent (superlative)

Captain Herschel expresses doubts of this, but it is evident that as the cusps closed both must have been within the space subtendent by half the length of the slit.

1875, Lieutenant-Colonel J.F. Tennant, “Report on Observations of the Total Elipse of the Sun”, in Memoirs of the Royal Astronomical Society, volume 42, page 9:

Make subtendent lines to them, and cut away the arches; then erect plain Looking-glasses, that may be of the same latitude, and of the same parallel lines, and the same longitude; glew them fast together, and fit them lo, that they may not be pulled asunder, as they are joyned long-ways, and erected upon a plain superficies.

1957, Giambattista della Porta, Natural Magick, page 359:

An outermost, “ballooning” membrane was seen to form dilated vesicles above the subtendent “armour” layer.

1988, British Phycological Journal, page 53:

(geometry) That subtends; that lies opposite to a specified angle or point.


Sylvester-Gallai theorem

Named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.

(geometry) A theorem stating that, given a finite number of points in the Euclidean plane, either all the points lie on a single line, or there is at least one line which contains exactly two of the points.

James Joseph Sylvester
Sylvester-Gallai theorem
Tibor Gallai


de Bruijn-Erdős theorem

Named after the mathematicians Nicolaas Govert de Bruijn and Paul Erdős.

(geometry) A theorem that states a lower bound on the number of lines determined by n points in a projective plane.


Pasch's axiom

Its essential role was discovered in 1882 by the German mathematician Moritz Pasch.

(geometry) A statement in plane geometry, used implicitly by Euclid, which cannot be derived from Euclid's postulates. It states that, if a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.


Pasch's theorem

Stated in 1882 by the German mathematician Moritz Pasch.

(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).


vertex angle

(geometry) An angle associated with a vertex of an n-dimensional polytope.


orthocentral

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cutplane

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(geometry) A plane that intersects a solid shape


bifolium

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Cartesian coordinate
Cartesian coordinate

(geometry) A quartic plane curve whose equation in Cartesian coordinates is (x²+y²)²=ax²ʸ.


heptadecahedron

en-noun

heptadecahedron (plural heptadecahedra or heptadecahedrons)

heptadecahedra (plural)
heptadecahedrons (plural)

(geometry) A polyhedron with seventeen faces.


pentadecahedron

pentadecahedron (plural pentadecahedra or pentadecahedrons)

pentadecahedra (plural)
pentadecahedrons (plural)

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(geometry) A polyhedron with fifteen faces.


hexadecahedron

hexadecahedron (plural hexadecahedra or hexadecahedrons)

hexadecahedra (plural)
hexadecahedrons (plural)

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(geometry) A polyhedron with sixteen faces.


enneadecahedron

enneadecahedron (plural enneadecahedra or enneadecahedrons)

enneadecahedra (plural)
enneadecahedrons (plural)

Guy [24] and Knowlton have independently discovered a uniform unistable enneadecahedron.

Klamkin noted that [1] contains an improved solution by R.K. Guy to the polyhedron problem, consisting of a doubly truncated 17-sided convex prism, making it an enneadecahedron (19 faces). […] Ogilvy referred to his book [4], where he describes Guy's enneadecahedron.

1980, Crux Mathematicorum, pages 28, 29:

Our solid is a 17-sided prism, half of whose section is illustrated in Fig. 5, truncated obliquely as in the side view of Fig. 6, so it is an enneadecahedron.

1990, Murray S. Klamkin, Problems in Applied Mathematics: Selections from SIAM Review, Society for Industrial and Applied Mathematics, →ISBN, page 478:

(geometry) A polyhedron with nineteen faces.


rhombi-

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rhombidodecahedron

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(geometry) A nonconvex uniform polyhedron with square faces and sharing symmetry of a dodecahedron.


Gougu

Transliteration of Chinese 勾股 (gōugǔ), literally the longer and shorter legs of a right triangle.


Gougu theorem

Borrowed from Chinese 勾股定理 (gōugǔ dìnglǐ), literally the longer and shorter legs of a right triangle theorem.


apeirotope

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(geometry) A generalized polytope having infinitely many facets.


dihectagon

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Dihectagon — 200 Sides

2014, Charles W. Nigh, Geometric Shapes & Forms:

A Dihectagon is a 200 sided shape. Made using the polygon tool.

2016 February 29, Greg N., “Dihectagon”, in 3D Warehouse:

Figure 4.57 A dihectagon looks very similar to a circle. Figure 4.57 shows a 200-sided shape (a dihectagon). It looks very much like a circle now, but it isn't

2017, Ed Southall, Yes, But Why? Teaching for Understanding in Mathematics, page 90:

I understand what a hectogon (a one hundred-sided figure) is for instance, but cannot imagine one, at least not in such a way that my image would be distinct from my image of a dihectagon (see Descartes 2008 [1641]: 51-52).

2019, Kevin Lynch, “Knowing What an Experience Is Like and the Reductive Theory of Knowledge‐wh”, in Analytic Philosophy, volume 61, number 3, →DOI:

(geometry, rare) A polygon with two hundred sides and two hundred angles.

200-угольник

двухсотугольник


triacontadigon

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(geometry) A polygon with thirty-two sides and thirty-two angles.


tetracontadigon

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Construct an edge-to-edge tessellation with a triangle, a heptagon, and a tetracontadigon. Is it possible to continue such a tessellation to have the full petal around the tetracontadigon?

2010, Sergei Abramovich, Topics in Mathematics for Elementary Teachers, page 90:

[…] (edge-to-edge tessellation with triangle, heptagon, and tetracontadigon, see Fig. 25). Fig. 25 Edge-to-edge tessellation with triangle, heptagon, and tetracontagon

2021, Sergei Abramovich, Michael L. Connell, Developing Deep Knowledge in Middle School Mathematics:

Using GeoGebra, construct an edge-to-edge tessellation with triangle, heptagon, and tetracontadigon. Is it possible to continue such a tessellation to have the full petal around the tetracontadigon?

2024, Sergei Abramovich, Developing Deep Knowledge in Middle School Mathematics, page 82:

(geometry) A polygon with forty two sides and forty two angles.

42-угольник

сорокадвухуго́льник


octadecahedron

octadecahedron (plural octadecahedra or octadecahedrons)

octadecahedra (plural)
octadecahedrons (plural)

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The cell structure was considered to consist of spherical holes but for the purposes of calculation, the cells were assumed to be octadecahedra. ln this model, a compressive force along the principal axis of the octadecahedron and parallel to.

1974, International Polymer Science and Technology, page 202:

In fact the structure of 2 can be related to the octadecahedron and the pentacapped trigonal prism in an interesting fashion

(geometry) A polyhedron with eighteen faces.


enneacontahexagon

Eventually, he worked his way up to a ninety-six-sided figure, an enneacontahexagon.

(geometry) A polygon with ninety six sides and ninety six angles.


icosikaidigon

The exterior wall is a 22-sided polygon (an icosikaidigon) and contains three large double-hung windows per bay.

2005, Baltimore Civil Engineering History, page 209:

A 22—sided shape once came up in relation to Activity 10.17, and various internet sources suggested we could call it an icosikaidigon!

2013, Essential Primary Mathematics, page 269:

(geometry) A polygon with twenty two sides and twenty two angles.


icosidigon

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tetracontakaihexagon

For example, a 46-sided polygon could be called a tetracontakaihexagon, or a combination of “tetracontagon” “kai” (often used in combining names) “hexagon.”

2012, Patricia Barnes-Svarney, Thomas E Svarney, The Handy Math Answer Book, page 184:

(geometry) A polygon with forty six sides and forty six angles.


icosagonal

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(geometry) Having the form of a icosagon


triacontakaitetragon

For instance a 34-sided polygon is called a triacontakaitetragon, and a 52-sided polygon is called a pentacontakaidigon.

2013, Mark A. Freitag, Mathematics for Elementary School Teachers, page 586:

(geometry) A polygon with thirty four sides and thirty four angles.


triacontatetragon

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linear triplet

(geometry) The three supplementary adjacent angles formed by three intersecting lines


equidiagonal

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(geometry) Of a convex quadrilateral: having the two diagonals of equal length.


Ford circle

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Every Ford circle is tangent to the horizontal axis y#61;0, and any two Ford circles are either disjoint or meet at a tangent.

There is a unique Ford circle associated with every rational number. Additionally, the axis y#61;0 can be considered a Ford circle with infinite radius, corresponding to the case p#61;1,#92;q#61;0.

Thus the Ford circle [1], drawn tangent to the real axis at #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, and having radius #92;textstylea#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125;, must contain in its interior some points belonging to #92;textstyle#123;p#95;n#125;#47;#123;q#95;n#125;, such as #92;textstyle#92;theta#43;ia#95;#123;n#43;1#125;#123;-1#125;q#95;n#123;-2#125; whose imaginary part lies between #92;textstyleq#95;n#123;-2#125; and #92;textstyleq#95;#123;n#43;1#125;#123;-2#125;.

1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:

A Farey fraction #92;textstyle#92;frackc defines a Ford circle #92;textstyle#92;mathcal#123;C#125;(k,c) in #92;textstyle#92;mathbbC. Its center is given by #92;textstylekc#43;i#92;frac 1#123;2c²#125; and its radius is #92;textstyle#92;frac1#123;2c²#125;. Two Ford circles #92;textstyle#92;mathcalC(a,b) and #92;textstyle#92;mathcalC(c,d) are tangent whenever #92;textstylead-bc#61;#92;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #92;textstyleF#95;N.

2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

Ford circles are a collection of horocycles in #92;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#95;#123;a#47;b#125; and C#95;#123;c#47;d#125; are tangent if and only if #92;vertad-bc#92;vert#61;1.

2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:

(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Ford-Kreis

any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle


absolute geometry

absolute geometry (plural absolute geometries)

From 1832; introduced by Hungarian mathematician János Bolyai (1802—1860).

絶対幾何学

1993 [Princeton University Press], Donald M. Davis, The Nature and Power of Mathematics, 2004, Dover, page 85,
Recall that absolute geometry is the set of statements that can be deduced from Euclid's first four postulates. Then existence of parallel lines is certainly a theorem of absolute geometry, while the question being addressed by most of the mathematicians discussed in this section is whether uniqueness of parallels is also a theorem of absolute geometry.

(geometry) The single (up to logical equivalence) geometry whose axiomatic system is equivalent to that of Euclidean geometry without the parallel postulate or any alternative.

1970, J. F. Rigby, Axioms for Absolute Geometry, III, Canadian Journal of Mathematics, Vol. XXII, No. 1, Canadian Mathematical Congress, University of Toronto Press, page 185,
A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper.

(geometry) Any geometry whose axiomatic system extends that of absolute geometry (in the singular sense) and neither assumes nor contradicts the parallel postulate.


birational geometry

Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.

1999, Yongbin Ruan, Surgery, Quantum Cohomology and Birational Geometry, Ya. Eliashberg, D. Fuchs, T. Ratiu, Alan Weinstein (editors), Northern California Symplectic Geometry Seminar, American Mathematical Society, Translations, Series 2, Volume 196, page 183,
Recently, some amazing relations between quantum cohomology and birational geometry have been discovered.

2008, Paltin Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, Ciro Ciliberto, et al., Projective Varieties with Unexpected Properties, Walter de Gruyter, page 317,
This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ℙⁿ (quasi-lines).

2009, Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Springer (Birkhäuser), Progress in Mathematics 276, page 233,
In this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform.

(algebraic geometry) A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets.


combinatorial geometry

combinatorial geometry (countable and uncountable, plural combinatorial geometries)

The generation of combinatorial types of point configurations and hyperplane arrangements point configurations and hyperplane arrangements has long been an outstanding problem of combinatorial geometry.

2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer,, page 425:

The following problem of Erdős [Er46] is possibly the best known (and simplest to explain) problem in combinatorial geometry. How often can the same distance occur among n points in the plane?

2006, Peter Brass, William O. J. Moser, János Pach, Research Problems in Discrete Geometry, Springer, page 183:

2012, Mohammed Mostefa Mesmmoudi, et al., Discrete Curvature Estimation Methods for Triangulated Surfaces, Ullrich Köthe, Annick Montanvert, Pierre Soille (editors), Applications of Discrete Geometry and Mathematical Morphology, Springer, LNCS 7346, page 28,
In combinatorial geometry, the most common discrete representation for a surface is a triangle mesh.

(geometry, uncountable) The field of mathematics which examines extremal problems of a combinatorial nature expressed geometrically.

For the generation of these combinatorial types no direct method is known, and it appears to be necessary to use combinatorial abstractions — allowable sequences of permutations, #92;lambda-functions, chirotopes, combinatorial geometries, or oriented matroids; in our work we will use oriented matroids [BLVS⁺99].

2003, Lukas Finschi, Komei Fukuda, “Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements”, in Boris Aronov, Saugata Basu, Janos Pach, Micha Sharir, editors, Discrete and Computational Geometry, Springer, page 425:

Our purpose in this chapter is to derive fundamental properties of combinatorial geometries, and to show how these properties strengthen our intuitive understandings of figures.[…]In this section, we give an axiom system for combinatorial geometries and then prove that each combinatorial figure is a combinatorial geometry.

2012, Don Row, Talmage James Reid, Geometry, Perspective Drawing, and Mechanisms, World Scientific, page 15:

(geometry, theory of matroids, countable) A simple matroid.


computational geometry

1991, Herbert Edelsbrunner, Lines in Space—A Collection of Results, Jacob E. Goodman, Richard D. Pollack, William L. Steiger (editors), Discrete and Computational Geometry: Papers from the DIMACS Special Year, American Mathematical Society, Association for Computing Machinery, page 77,
In computational geometry it is a truism that problems in three-dimensional space, ℝ³, are significantly more difficult and complex than the corresponding problems in the two-dimensional plane, ℝ².

2003, Ren-Hong Wang, Multivariate Spline and Geometry, Ren-Hong Wang (editor), Computational Geometry, American Mathematical Society, International Press, page 195,
In this survey, we will be concerned with the relationship between multivariate splines and geometries including computational geometry, differential geometry, and algebraic geometry.

Computational geometry processing is about practical algorithms that operate on geometric data sets, and these data sets tend to be rather large if they are to be useful.

2012, J. Andreas Bærentzen, Jens Gravesen, François Anton, Henrik Aanæs, Guide to Computational Geometry Processing, Springer, page 1:

(computer science, geometry) A field of study, within computer science, devoted to algorithms that solve problems formulated in geometric terms.


conformal geometry

conformal geometry (countable and uncountable, plural conformal geometries)

2000, Boris N. Apanasov, Conformal Geometry of Discrete Groups and Manifolds, Walther de Gruyter, page vii,
Generally, conformal geometries naturally appear at infinity for negatively curved Riemannian geometries. Moreover, due to M. Gromov's [5, 6] geometric approach to infinite groups, conformal geometry invents new fruitful methods in combinatorial group theory.

The lectures concerned the study of some non-linear partial differential equations related to curvature invariants in conformal geometry.

2004, Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry, European Mathematical Society, page 2:

This chapter gives a brief introduction of the basic theoretic foundations and profound and beautiful structures of conformal geometry.

2018, Miao Jin, Xianfeng Gu, Ying He, Yalin Wang, Conformal Geometry: Computational Algorithms and Engineering Applications, Springer, page 1:

conformal transformation
conformal transformation

(geometry) The study of conformal (angle-preserving) transformations on a space;

The study of conformal (angle-preserving) transformations on a space;
(formally) an ordered pair comprising a space and the group of conformal transformations that act on the space.