nonsqueezing

Relying on Gromov's nonsqueezing theorem for finite dimensional Hamiltonian systems, the argument is to approximate the solutions to the original infinite dimensional Hamiltonian system by a frequency truncated finite dimensional system, and then the nonsqueezing property is transferred to the infinite dimensional system.

2015, Sunghyun Hong, Soonsik Kwon, “Nonsqueezing property of the coupled KdV type system without Miura transform”, in arXiv:

(geometry) Being or relating to the theorem in symplectic geometry stating that one cannot embed a sphere into a cylinder via a symplectic map unless the radius of the sphere is less than or equal to the radius of the cylinder.


Hilbert curve

First described by the German mathematician David Hilbert in 1891.

(geometry) A continuous fractal space-filling curve.


hyperball

We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings..

2015, Jenő Szirmai, “Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings”, in arXiv:

(geometry) The set of all points that make a hypersphere.


hyperbolic geometry

en-noun

hyperbolic geometry (countable and uncountable, plural hyperbolic geometries)

Beltrami–Klein model
Felix Klein
Poincaré disk

The terminology was introduced in 1871 by Felix Klein, who classified some non-Euclidean geometries as hyperbolisch (“hyperbolic”), elliptisch (“elliptical”), and parabolisch (“parabolic”).

parallel postulate
parallel postulate#English

(geometry) A type of geometry that rejects the parallel postulate. Given a straight line L and a point P not on the line, more than one straight line can be drawn through P without intersecting L.


spatulous

en-adj

spatulous (comparative more spatulous, superlative most spatulous)

more spatulous (comparative)
most spatulous (superlative)

LL-Q1860 (eng)-I learned some phrases-spatulous.wav

From spatula + -ous (suffix forming adjectives denoting presence of a quality, usually in abundance).

suffix

[T]he ſhell of teredo is diſtinguiſhed as being tubular, cylindrical, and open at both ends, lower orifice furniſhed with two lozenge-ſhaped valves, and the upper with two ſpatulous opercules.

1819, Abraham Rees, “CONCHOLOGY”, in The Cyclopædia; or, Universal Dictionary of Arts, Sciences, and Literature. [...] In Thirty-nine Volumes, volume IX, London: Printed for Longman, Hurst, Rees, Orme, & Brown, […] [et al.], →OCLC, column 2:

The Nova Scotian Giantess.—A correspondent had the opportunity of inspecting this remarkably large female [Anna Swan] during the exhibition at the Egyptian Hall, and forwards the following notes:— [...] The hand is spatulous; the fingers short and subequal in length.

1869 April, “Anthropological News”, in The Anthropological Review, volume VII, number XXV, London: Asher & Co., […], →OCLC, page 215:

[page 10] Spatulous fingers, by which I mean those somewhat splay shaped, devote themselves to the useful, especially the physically useful, such as agriculture. [...] [page 11] Such fingers have superabundant energy, and though at first sight war appears utterly opposed to the qualities belonging to this spatulous type, it often affords an outlet to a craving for action and excitement which must find vent.

1886, Eveline Michell Farwell, “Square and Spatulous Tips”, in Fingers and Fortune: A Guide-book to Palmistry, 2nd edition, London: David Stott, […], →OCLC, pages 10–11:

And he would turn to Miss Moorsom for approval, lowering protectingly his spatulous nose and looking up with feeling from under his absurd eyebrows, which grew thin, in the manner of canebrakes, out of his spongy skin.

1913 December, Joseph Conrad, “The Planter of Malata”, in Within the Tides: Tales, London, Toronto, Ont.: J[oseph] M[alaby] Dent & Sons Ltd., published 1915, →OCLC, chapter IV, page 44:

Braking mechanism for the front wheels of motor cars comprising a bearing cap mounted on the stub-axle of the car, a double lever pivotally mounted in said bearing cap one end of said lever being of spatulous shape, [...]

1921 July 25, Marius Adenot, Braking Mechanism for the Front Wheels of Motor Cars, US Patent 1,437,080, column 2, lines 59–64:

A group of syringoceles may open into the urethra by a spatulous ostium; these are classified as perforated syringoceles.

2005, W. van’t Hoff, P. G. Duffy, “Urolithiasis”, in David M. Burge, D. Mervyn Griffiths, Henrik A. Steinbrecher, Robert A. Wheeler, editors, Paediatric Surgery, 2nd edition, London: Hodder Arnold; Boca Raton, Fla.: CRC Press, →ISBN, part 6 (Urology), page 481, column 2:

Resembling a spatula in shape; having a round, flat tip.


figure of eight

en-noun

figure of eight (plural figures of eight)


GSO

geosynchronous orbit
geosynchronous orbit#English


tesseractic

(geometry) Relating to a tesseract; relating to four-dimensional space or to a 4-polytope


trisector

(geometry) A line or curve that trisects something.


starlikeness

(geometry) The property of being starlike.


rhombitrihexagonal

(geometry) Relating to a semiregular tiling of the Euclidean plane with one triangle, two squares, and one hexagon on each vertex.


isotoxal

(geometry, of a polytope or tiling) That has symmetries which act transitively on its edges.

edge-symmetric

having symmetries that act transitively on its edges


scalebound

From scale + bound, coined by the mathematician Benoit Mandelbrot in a 1981 paper.

(geometry) Of an object: for which characteristic elements of scale, such as length and width, are few in number and each with a clearly distinct size.


buckyonion

Blend of buckminsterfullerene + onion, modelled on earlier buckyball, with reference to the concentric layers of an onion.

Detections of C60 and C70 fullerenes in planetary nebulae (PNe) of the Magellanic Clouds and of our own Galaxy have raised the idea that other forms of carbon such as hydrogenated fullerenes (fulleranes like C60H36 and C60H18), buckyonions, and carbon nanotubes, may be widespread in the Universe.

2016, J. J. Díaz-Luis, D. A. García-Hernández, A. Manchado, F. Cataldo, “A search for hydrogenated fullerenes in fullerene-containing planetary nebulae”, in arXiv:

(chemistry) A concentric multishell fullerene structure.


apeirogonal

(geometry) Having the form of an apeirogon.


flowsnake

From flow + snake, describing its movements, and as a spoonerism for snowflake.


Gosper curve

(computer graphics) A particular fractal object, a space-filling curve that can tile the plane.


V-shaped

Shaped like the letter V; especially, designating one of the two major types of valley (the other being U-shaped), typically formed by flowing water.

A-shaped - B-shaped - C-shaped - D-shaped - E-shaped - F-shaped - G-shaped - H-shaped - I-shaped - J-shaped - K-shaped - L-shaped - M-shaped - N-shaped - O-shaped - P-shaped - Q-shaped - R-shaped - S-shaped - T-shaped - U-shaped - V-shaped - W-shaped - X-shaped - Y-shaped - Z-shaped


bidisc

(mathematics) A polydisc of the second order


circumscribed circle

circumscribed circle (plural circumscribed circles)


inscribed circle

inscribed circle (plural inscribed circles)


point of tangency

point of tangency (plural points of tangency)

(geometry) The point where a tangent line and its curve meet

切點

the point where a tangent line and its curve meet


Coxeter group

Named for British-born Canadian mathematician H. S. M. Coxeter (1907–2003).

We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. de Clerck, editors, Finite Geometries and Combinatorics, page 277:

The above definitions, of Cayley graph, reflection, wall, half-space, folding, Bruhat order, etc., work for any Coxeter group.

2007, Michael B. Smythe, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem, editors, Handbook of Spatial Logics, page 795:

The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n,1).

2012, Daniel Allcock, “The Reflective Lorentzian Lattices of Rank 3”, in Memoirs of the American Mathematical Society, number 1033, page vii:

(mathematics, geometry, group theory) Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics.

Coxeter group
Harold Scott MacDonald Coxeter


vertex figure

1970, Harold Scott Macdonald Coxeter, Twisted Honeycombs, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics: Number 4, American Mathematical Society, page 14,
Thus the vertex figure of a polygon is the line-segment whose end-points are the vertex figures of the two edges at one vertex, the vertex figure of a polyhedron or tessellation {p, q} is the polygon {q} whose edges are the vertex figures of the q faces at one vertex, and the analogous "vertex figure" of the honeycomb of cubes (§1) is the polyhedron whose faces are the vertex figures of the eight cubic cells that come together at one vertex.

Now try to see which of these vertex figures cannot be continued to give a semiregular tiling of the whole plane.

1987, A. Gardiner, Mathematical Puzzling, Dover, published 1999, page 137:

At each vertex of a tessellation, the vertex figure is the polygon formed by joining in order the mid-points of all edges incident with the vertex. A tessellation of a surface for which all the participating polygons and vertex figures are regular and equal is termed a regular tessellation.

2004, Michael F. Worboys, Matt Duckham, GIS: A Computing Perspective, 2nd edition, page 188:

(geometry) The figure exposed when a corner of a polyhedron or polytope is sliced off.


oloid

Furthermore, we derive the expectations of the mean width, the surface area and the volume of the intersections of a fixed oloid and a moving ball, as well as of a fixed and a moving oloid.

2016, Uwe Bäsel, “The mean width of the oloid and integral geometric applications of it”, in arXiv:

(geometry) A three-dimensional curved geometric object, the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the other circle.


quasicircle

We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space, which also enable us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in Douglas-Osborn space on quasicircles.

2016, Eric Schippers, Wolfgang Staubach, “Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks”, in arXiv:

(mathematics) A Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself.


spirallikeness

(geometry) The property of being spirallike.


deorbit burn

(astronautics) A firing of the engines of a spacecraft to enable it to re-enter the Earth's atmosphere.


rhombic dodecahedron

rhombic dodecahedron (plural rhombic dodecahedra or rhombic dodecahedrons)

rhombic dodecahedra (plural)
rhombic dodecahedrons (plural)

Doing this we would find that the network which is dual to the space filling array of rhombic dodecahedra is the space filling array of tetrahedra and octahedra in which 12 edges meet at the vertices which fall at the centers of the original rhombic dodecahedra.

1990, [1978], Peter Pearce, Structure in Nature Is a Strategy for Design, page 5:

Crystallographers are familiar with the rhombic dodecahedron as a domain of reference with which to account for the growth and structure of natural crystals.

2007, Scott Eastham, American Dreamer: Bucky Fuller and the Sacred Geometry of Nature, page 160:

Rhombic Dodecahedrons appear as well in the unit cells of diamonds, four vertices absent with the chemical bonds on the remaining edges. The Rhombic Dodecahedron can be used to tessellate 3D space in a manner similar to the Cube.

2013, L. Dennis, Brender McNair, N. J. Woolf, L. H. Kauffman, “6: The Mereon 120/80 — Form Informing Function”, in Lynnclaire Dennis, Jytte Brender McNair, Louis H. Kauffman, editors, The Mereon Matrix: Unity, Perspective and Paradox, page 135:

(geometry) A convex polyhedron that has 12 congruent rhombic faces, 24 edges and 14 vertices of two types (eight 3-edge and six 4-edge) and is a Catalan solid.


Schläfli symbol

Schläfli symbol (plural Schläfli symbols)

From Schläfli (“a surname”) + symbol, after 19th-century Swiss mathematician Ludwig Schläfli (1814–1895).

1995, Unnamed translator, E. B. Vinberg, Polynomial Group, article in M. Hazewinkel (editor), Encyclopaedia of Mathematics, Volume 4: Monge–Ampère Equation — Rings and Algebras, 2013, International Edition, page 481,
The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — {3, 3}, the cube — {4, 3}, the octahedron — {3, 4} — the dodecahedron — {5, 3}, and the icosahedron — {3, 5}.

For the regular tiling of squares the Schläfli symbol is {4,4}, indicating that there are four squares surrounding each vertex. And for the regular tiling of hexagons the Schläfli symbol is {6,3}, indicating that there are three hexagons surrounding each vertex.

2007, Michael E. Mortenson, chapter L, in Geometric Transformations for 3D Modeling, page 248:

The 24-cell has has 24 octahedral cells, 24 vertices, 96 triangular faces and 96 edges. Its Schläfli symbol is {3, 4, 3} from which we see that the vertex figure is a cube.

2012, John Barnes, Gems of Geometry, 2nd edition, page 79:

(geometry) A notation that recursively encodes certain properties of a specified regular polytope or tessellation.

Schläflin symboli

notation encoding certain properties of a regular polytope or tessellation


Perhaps the most obvious next step is to investigate 4-polytopes with respect to symmetry equivalence.

1984, London Mathematical Society lecture note series, numbers 88-90, page 101:

The situation for 4-polytopes is fundamentally different from that for 3-dimensional polytopes. One reason is that there is no similar reduction of 4-polytope theory to a combinatorial (graph) problem.

2004, Martin Henk, Jürgen Richter-Gebert, Günter M. Ziegler, “16: Basic properties of convex polytopes”, in Joseph O'Rourke, Jacob E. Goodman, editors, Handbook of Discrete and Computational Geometry, 2nd edition, page 369:

For d = 4, Hamiltonian cycles in the regular 4-polytopes are known to exist. However, it seems that so far no decision about the existence or non-existence of 1-Hamiltonian surfaces in the 2-skeleton of any of the three sporadic regular 4-polytopes can be made, compare [120].

2011, Felix Effenberger, Hamiltonian Submanifolds of Regular Polytopes, page 40:


Schlegel diagram

Schlegel diagram (plural Schlegel diagrams)

From Schlegel (“a surname”) + diagram, after German mathematician Victor Schlegel, who introduced the diagram in 1886.

The location of the point x₀ can always be chosen so that the edges in the Schlegel diagram can be drawn as non-intersecting straight lines.

1999, R. B. King, “1: Topology in Chemistry”, in D Bonchev, D.H Rouvray, editors, Chemical Topology: Introduction and Fundamentals, page 21:

Schlegel diagrams are a useful way to explore how these polyhedra can be linked (Hoppe and Köhler 1988).

2002, Ian David Brown, chapter L, in The Chemical Bond in Inorganic Chemistry: The Bond Valence, page 150:

These structures may all be represented on a planar surface by their Schlegel diagrams. A polyhedron Schlegel diagram is its networks of edges and vertices drawn in a special way: if you hold the polyhedron so close to your face that one of faces frames the entire polyhedron and you see all the other edges meeting inside that frame, then you have a Schlegel diagram.

2013, Arthur L. Loeb, “5: Polyhedra: Surfaces or Solids?”, in Marjorie Senechal, editor, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, page 69:

(geometry) A projection of a polytope from n-dimensional space to n-1 dimensions through a point beyond one of its faces; especially such a projection (itself represented in 2 dimensions) of a 3- or 4-dimensional polytope.


tetrakis hexahedron

tetrakis hexahedron (plural tetrakis hexahedra or tetrakis hexahedrons)

tetrakis hexahedra (plural)
tetrakis hexahedrons (plural)

Of the fourteen crystals examined, five were from the Karnul district (four tetrakis-hexahedra and one octahedron with tetrakis-hexahedron), one from Sambalpur (tetrakis-hexahedron with octahedron, four from Panna (much distorted tetrakis-hexahedra), the remaining four said to have come from Simla.

1904, Max Bauer, translated by L. J. Spencer, Precious Stones, volume 1, published 1968, page 154:

In the Isometric Tribe can be found the families of the Tetrakis Hexahedrons and Octahedrons, with their various descendants.

1909, Marshman Edward Wadsworth, Crystallography: An Elementary Manual for the Laboratory, page 189:

The goniometrical angle over the longer edges, that is, the angle between the normals to each pair of faces intersecting in a longer edge, in the cases of the two tetrakis hexahedra most frequently met with, has the following values:

1922, Alfred Edwin Howard Tutton, Crystallography and Practical Crystal Measurement, volume 1, page 153:

Not one exhibits plane faces or straight edges, being mostly distorted triakis-octahedra or tetrakis-hexahedra with finely striated curved faces.

1946, Mineralogical Society of America, Mineralogical Abstracts, volume 9, page 109:

Because of this another polyhedron – the tetrakis hexahedron – that has 14 vertices and 24 faces might be also a good model. The tetrakis hexahedron is slightly less regular than an icosahedron.

2010, Sumudu P. Leelananda, Yaping Feng, Pawel Gniewek, Andrzej Kloczkowski, Robert L. Jernigan, “6: Statistical Contact Potentials in Protein Coarse-Grained Modeling: From Pair to Multi-body Potentials”, in Andrzej Kolinski, editor, Multiscale Approaches to Protein Modeling, page 150:

(geometry) A Catalan solid that has 24 congruent triangular faces and 36 edges and can be imagined as a cube with a square pyramid on each face.

tetracisesaedro

polyhedron with 24 triangular faces and 36 edges


angular defect

Prop. XVI. If the angular defects of two triangles are equal, the areas of the triangles are equal.

1864, Transactions of the Royal Society of Edinburgh, volume 23, page 444:

We have seen that under any reasonable definition of area, the angular defect of a triangle is proportional to its area.

1995, Arlan Ramsay, Robert D. Richtmyer, Introduction to Hyperbolic Geometry, page 120:

In hyperbolic geometry, similar triangles do not exist and the area of a triangle is directly proportional to its angular defect[…].

1998, Kappa Mu Epsilon, The Pentagon: A Mathematics Magazine for Students, page 7:

(geometry, non-Euclidean geometry) The amount by which the total of the interior angles of a triangle is less than 180° (π radians); the amount by which the total of the internal angles of a polygon is less than what would be expected on the Euclidean plane.

angular deficiency

amount by which the total of the interior angles of a polygon is less than on the Euclidean plane

The angular defect clearly has something to do with curvature, because the larger it is, the more pointed the surface is at the vertex.

2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, page 604:

2014, C. R. Calladine, The Static-geometric Analogy in the Equations of Thin Shell Structures, W. Olszak, Thin Shell Theory: New Trends and Applications, page 294,
Figure 4(c) shows a flattened view of a small part of the undeformed polygonalized S-surface, consisting of the triangles surrounding a particular vertex. When the S-surface is strained there will be a consequent change of angular defect, which we wish to calculate.

2015, Jan Guichelaar (translator and editor), Alex Van Den Brandhof, Arnout Jaspers (editors), Half a Century of Pythagoras Magazine, page 164,
Theorem. For a spherical polyhedron the total angular defect equals 720°.

(geometry) The amount by which the total of the angles around a vertex of a polyhedron is less than 360° (2π radians).


symmetry group

[The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
In this chapter, we study symmetry groups of fulleroids.

2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

(geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

full symmetry group

emphasises the inclusion of orientation-reversing transformations


second cosmic velocity

From second + cosmic + velocity, from being the next threshold velocity greater than the first cosmic velocity.

Escape velocity of Earth (interplanetary speed).


second cosmic velocity

second cosmic velocity (plural second cosmic velocities)

(by extension) Escape velocity of the home body.


first cosmic velocity

From first + cosmic + velocity, from being the first velocity greater than that able to be held upon the Earth, thus entering the realm of the Cosmos.

The velocity in a circular orbit about the Earth.


third cosmic velocity

From third + cosmic + velocity, from being the next threshold velocity greater than the second cosmic velocity.

The escape velocity of the Solar System (interstellar speed).


third cosmic velocity

third cosmic velocity (plural third cosmic velocities)

(by extension) The escape velocity of a star system.


cross-polytope

The case p = 1, which is actually important to us, is illustrated in Figure 1.14; we are looking at the intersection of a cross-polytope with the probability simplex.

2006, Ingemar Bengtsson, Karol Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, page 25:

2008, Robert Erdahl, Andrei Ordine, Konstantin Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices, Matthias Beck, et al. (editors), Contemporary Mathematics 452: Integer Points in Polyhedra — Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS-IMS-SIAM Joint Summer Conference, June 11-15, 2006, page 104,
The convex hulls of such crosses often appear as cells in Delaunay tilings – cross polytopes are examples, as are the more spectacular symmetric perfect Delaunay polytopes.

The cross polytope βₙ is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because its vertices are located equidistant from the origin, along the Cartesian axes in n-space. The cross polytope in n dimensions is bounded by 2ⁿ (n - 1)-simplices, has 2n vertices and 2ⁿ(n - 1) edges.

2009, Herbert W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer, page 263:

If one works with the l₁ norm, a ball (the set of all points whose distance lies within a certain radius around a point of interest) has the shape of a cross-polytope. A one-dimensional cross-polytope is a line segment, a two-dimensional cross-polytope is a square, for three dimensions, an octahedron, and so forth.

2010, Rainer Typke, Agatha Walczak-Typke, “Indexing Techniques for Non-metric Music Dissimilarity Measures”, in Zbigniew W. Raś, Alicja A. Wieczorkowska, editors, Advances in Music Information Retrieval, Springer, page 10:

While centrally symmetric d-polytopes with 2d vertices are affinely equivalent to the d-dimensional cross-polytope, a complete classification of the centrally symmetric d-polytopes with 2d + 2 vertices is out of reach.

2003, Branko Grünbaum, Convex Polytopes, 2nd edition, Springer, page 121b:

(geometry) A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex.


Coxeter diagram

Coxeter diagram (plural Coxeter diagrams)

It suffices to make use of the tables above and observe that the Coxeter diagram of a direct sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands.
For the description of C⁻-matrices satisfying condition (2) we shall use Coxeter diagrams with the additional stipulation that in case aᵢⱼ < −1 the vertices vᵢ and vⱼ are joined by a dotted line with index −aᵢⱼ. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean the Coxeter diagram of the corresponding C⁻-matrix.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, page 358:

An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. Buekenhout, J. Doyen, F. de Clerck, J. A. Thas, J. W. P. Hirschfeld, editors, Finite Geometries and Combinatorics, 2nd International Conference, page 277:

The Coxeter diagrams of type E₇ and E₈ have no nontrivial automorphisms, so σ₀ is trivial in those cases.[…]The Coxeter diagram of type F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have any nontrivial automorphisms.

2008, Peter Abramenko, Kenneth S. Brown, Buildings: Theory and Applications, page 259:

Coxeter-Dynkin diagram
Coxeter-Dynkin diagram

(geometry, algebra) A Coxeter-Dynkin diagram.


hyperellipsoid

(geometry) A figure, related to the hypersphere, having ellipsoidal rather than spherical symmetry


hyperellipsoidal

A regularized Mahalanobis distance can be used for extracting hyperellipsoidal clusters [30 ].

2015 December 12, “A Comparison Study on Similarity and Dissimilarity Measures in Clustering Continuous Data”, in PLOS ONE, →DOI:

(geometry) Relating to, or having the form of a hyperellipsoid


snub disphenoid

snub disphenoid (plural snub disphenoids)

So named in 1966 by mathematician Norman Johnson in his classification of the Johnson solids. See also Snub disphenoid on Wikipedia.Wikipedia

For the snub disphenoid, our existence proof used the intermediate value theorem in the real numbers to argue that as AB decreases and CD increases there is a point where they become equal.

2005, Robin Hartshorne, Geometry: Euclid and Beyond, page 457:

In this structure-type, the eight-coordinate polyhedra (snub disphenoids) share edges to form a framework composed of cross-linked chains that extend along the [100] and [010] directions.

2006, Andrew J. Locock, “Chapter 6: Crystal chemistry of actinide phosphates and arsenates”, in Sergey Krivovichev, Peter Burns, Ivan Tananaev, editors, Structural Chemistry of Inorganic Actinide Compounds, page 225:

Thus, at a basic level this activity is about exploring such polyhedra, starting with the regular cases of the tetrahedron, octahedron, and icosahedron and moving into other solids like the triangular dipyramid and snub disphenoid.

2013, Thomas Hull, Project Origami: Activities for Exploring Mathematics, 2nd edition, page 153:

equilateral triangle
equilateral triangle

A non-regular convex polyhedron that has 12 equilateral triangles as faces and 18 edges and is a Johnson solid.

Norman Johnson (mathematician)
snub disphenoid

Siamese dodecahedron

polyhedron with 12 equilaterally triangular faces

deltahedron

polyhedron with faces that are all equilateral triangles


equitangential

(geometry) Of a curve: having tangential segments that are all of the same length.


triacontagon

From Ancient Greek τρῐᾱ́κοντᾰ (triā́konta) + -gon.

suffix

After projection onto the (x₁, x₂) plane, the A’s, B’s, C’s, and D’s form four concentric triacontagons, inscribed in circles of radii a, b, c, and d.

1947, H. S. M. Coxeter, Regular Polytopes, published 2012, unnumbered page:

1964 July, B. L. Chilton, On the Projection of the Regular Polytope {5, 3, 3} into a Regular Triacontagon, Canadian Mathematical Bulletin, Volume 7, Number 3, page 386,
In the projection the 600 points can be classified in 12 sets: each set consists of either of 30 or of 60 vertices which lie on a circle concentric with the outer triacontagon.

1997, Johannes Kepler, E. J. Aiton, Alistair Matheson Duncan, Judith Veronica Field (translators), The Harmony of the World, [1619, Harmonices Mundi], American Philosophical Society, Volume 309, page 59,
What should the decision now be about the sides of the Triacontagon?


deltoidal icositetrahedron

en-noun

deltoidal icositetrahedron (plural deltoidal icositetrahedra or deltoidal icositetrahedrons)

deltoidal icositetrahedra (plural)
deltoidal icositetrahedrons (plural)

These correspond to the foregoing U- and F-vertices of the deltoidal icositetrahedron whose three-edged vertices are "completely truncated"; that is to say, the truncating planes pass through the three end points of the edges radiating from the truncated vertex.

1958, The Mathematical Gazette, volume 42, page 20:

The mineral displays a well-developed habit of deltoidal icositetrahedra with dull faces.

1994, Proceedings of the Ninth Quadrennial IAGOD Symposium, International Association on the Genesis of Ore Deposits, page 593:

At the Olympus Mons Casino on Mar's they have craps games where the die are rhombic dodecahedron (12 sided), or tetrakis hexahedron (24 sides) or deltoidal icositetrahedron (24 sides). That's for your benefit, Gabby said to me.

2012, Ivan Pasztor, Of All The Things I've Lost, page 69:

(geometry) A convex polyhedron that has 24 congruent kite-shaped faces and 48 edges and is a Catalan solid, being the dual polyhedron of the rhombicuboctahedron.


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.


truncated tetrahedron

truncated tetrahedron (plural truncated tetrahedra or truncated tetrahedrons)

truncated tetrahedra (plural)
truncated tetrahedrons (plural)

The space of the El Al pyramid is created essentially by the packing of truncated tetrahedrons.

1968, Cindy Marriott, A Critical Analysis of Israeli Architecture, 1948-1967: A Study in Cultural Conflict, page 63:

The tendons now form the edges of a slightly distorted truncated tetrahedron.

1976, Anthony Pugh, An Introduction to Tensegrity, page 35:

This ideal net gives rise to the formation of B₁₂ truncated tetrahedra, where the A cations are lodged.

2000, Angel Vegas, Cations in Inorganic Solids: Crystallography Reviews, volume 7, number 3, page 248:

2009, Xiaotian Lin, Miao Jin, Feng Luo, Xiangfen David Gu, Discrete Curvature Flows for Surfaces and 3-Manifolds, Frank Nielsen (editor), Emerging Trends in Visual Computing: LIX Fall Colloquium, ETVC 2008, page 69,
Once the edge lengths of the tetrahedral mesh have been obtained, we can realize it in the hyperbolic space ℍ³ First, we introduce how to construct a single truncated tetrahedron; then we explain how to glue multiple truncated tetrahedra by hyperbolic rigid motion.

2010, K.M. Kosuda, J. M. Bingham, K. L. Wustholz, R. P. Van Dyne, 9: Nanostructures and Surface-Enhanced Raman Spectroscopy, Gary P. Wiederrecht (efitor), Handbook of Nanoscale Optics and Electronics, page 318,
Without any further modification, the shapes of the nanoparticles fabricated by NSL are truncated tetrahedrons. However, by thermal annealing the truncated tetrahedrons are transformed into hemispheres.

(geometry) A convex polyhedron, with four triangular and four hexagonal faces, that is an Archimedean solid and can be constructed by truncating a regular tetrahedron at each vertex by one-third the length of each adjoining edge.


snub dodecahedron

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

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#x3D;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#x3D;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


truncated icosidodecahedron

truncated icosidodecahedron (plural truncated icosidodecahedra or truncated icosidodecahedrons)

truncated icosidodecahedra (plural)
truncated icosidodecahedrons (plural)

LL-Q1860 (eng)-Flame, not lame-truncated icosidodecahedron.wav

From truncated + icosidodecahedron, referring to a partial construction method for the figure. (The resultant figure requires further distortion to make the faces regular.)

The Solid Shapes Lab, another Science Materials Center contribution to future mathematicians, together to form not only regular polyhedrons but rhombicosidodecahedrons, truncated icosidodecahedrons, and such.

1961, The New Yorker, Volume 37, Part 4, page 172:

The truncated icosahedron and truncated icosidodecahedron have fivefold symmetry and cannot four four-connected frameworks with lattice symmetry.

1962, The Mineralogical Magazine and Journal of the Mineralogical Society, volume 33, page 1008:

It is also a Catalan solid, the disdyakis triacontahedron, dual of the truncated icosidodecahedron.

2007, Numerical Analysis and Applied Mathematics: International Conference of Numerical Analysis and Applied Mathematics, page 95:

An Archimedean solid with 62 regular faces (12 decagons, 20 hexagons and 30 squares) and 180 edges.

icosidodecaedro troncato

polyhedron with 12 decagonal, 30 hexagonal and 30 square faces


midsphere

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


polygamma function

polygamma function (plural polygamma functions)

(mathematics) A meromorphic function on ℂ, defined as the (m+1)-th derivative (for a given order m) of the logarithm of the gamma function.


digamma function

digamma function (plural digamma functions)

(mathematics) The first of the polygamma functions, being the logarithmic derivative of the gamma function.


trigamma function

trigamma function (plural trigamma functions)

(mathematics) The second of the polygamma functions, after the digamma function.


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 #x5C;left#x5C;vertx-z#x5C;right#x5C;vert#x3D;#x5C;alpha#x5C;left#x5C;verty-z#x5C;right#x5C;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 #x5C;mathcal#x7B;l#x7D;#x5F;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 #x5C;zeta#x3D;1#x2F;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.


differential operator

differential operator (plural differential operators)

Define P(D) to be the differential operator given by P(D)#x3D;D²#x2B;5D#x2B;6, where D is the differentiation operator. Then P(D)(#x5C;sin(x))#x3D;-#x5C;sin(x)#x2B;5#x5C;cos(x)#x2B;6.

[…]these conditions appeared in the very first papers on ordinary differential operators.

2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:

A differential operator P#x5C;left(D#x5C;right) is hypoelliptic if for any domain ⊂ #x5C;mathbb#x7B;R#x7D;ⁿ any solution u of the equation P#x5C;left(D#x5C;right)u#x3D;0 from the class #x5C;mathfrak#x7B;D#x7D;'#x5C;left(#x5C;Omega#x5C;right) is a function from C#x5C;infty(ω) for any open set ω ⊂⊂ #x5C;Omega.
A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].

2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:

Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.

1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251:

mathematical analysis
mathematical analysis

(mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).


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

(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.


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.


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


midcircle

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

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).


plane geometry

The study of figures having parts that lie in a singular plane.


hypocube

A square, or any other geometric figure analogous to a cube but having fewer than three dimensions.


hexlet

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


Hohmann transfer orbit

Hohmann transfer orbit (plural Hohmann transfer orbits)

Named after Walter Hohmann, German scientist who described it in 1925.

(aerospace) An elliptical orbit used to transfer between two circular orbits of different radii in the same plane.


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.


colog

(mathematics) Abbreviation of cologarithm.


CHZ

Circumstellar habitable zone
w:Circumstellar habitable zone

Initialism of Circumstellar habitable zone.


cubinder

(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


Brillouin function

Brillouin function (plural Brillouin functions)

Named after the physicist Léon Brillouin (1889–1969).

statistical mechanics
statistical mechanics

A special function that appears when studying an idealized paramagnetic material in statistical mechanics.


horodisk

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space #x5C;mathbb#x7B;H#x7D;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 #x5C;Omega#x5C;subset#x5C;mathbb#x7B;H#x7D;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:


smooth function

smooth function (plural smooth functions)

Any ANALYTIC FUNCTION is smooth. But a smooth function is not necessarily analytic.

2003, Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, →ISBN, page 419:

(mathematics) a function that has derivatives of all finite orders everywhere in its domain.


hyperradius

hyperradius (plural hyperradii or hyperradiuses)

hyperradii (plural)
hyperradiuses (plural)


subface

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.


hypercuboid

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

nadkvádr

analogue of a cuboid in four or more dimensions


paracomplex

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

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


codifferential

One important concept is the introduction of codifferential operator

2015, 1:19:34 from the start, in 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):

differential geometry
differential geometry

(differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β).


functionhood

He did not distill the essential idea of functionhood: that to each x in the domain there corresponds a unique y in the range.

1999, William Dunham, Euler: The Master of Us All, The Mathematical Association of America, published 1999, page 17:

(mathematics, logic, grammar) The state or quality of being a function.


Ricci flow

(differential geometry) A process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.


plesiohedron

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.


gyrobifastigium

gyrobifastigium (plural gyrobifastigia or gyrobifastigiums)

gyrobifastigia (plural)
gyrobifastigiums (plural)

To elucidate this complex correlation, we carried out detailed Monte Carlo simulations of six convex space-filling polyhedrons, namely, truncated octahedrons, rhombic dodecahedrons, hexagonal prisms, cubes, gyrobifastigiums and triangular prisms.

2011 February, Umang Agarwal, Fernando A Escobedo, “Mesophase behaviour of polyhedral particles”, in Nature materials, volume 10:

They observed hard triangular prisms and hard gyrobifastigia to directly transform into their space-filling crystal, while hard truncated octahedra, rhombic.

2012, Amir Haji Akbari Balou, Thermodynamics of the Hard Tetrahedron System (PhD Dissertation in Chemical Engineering, University of Michigan):

But there are solids constructed from two or more regular prisms that also pack space. An interesting example (with an equally interesting name) is the gyrobifastigium, constructed from two regular triangular prisms. See Figure 1.2.14 for a picture of a gyrobifastigium, and a pattern (called a net) for constructing one.

2015, Claudi Alsina, Roger B. Nelsen, A Mathematical Space Odyssey: Solid Geometry in the 21st Century, →ISBN:

A polyhedral solid formed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is noted for being one of the few regular polyhedra that packs in three dimensions.

異相双三角柱

いそうそうさんかくちゅう


abracadabrangle

A representation of the word "abracadabra" in the visual form of a triangle, used as an amulet or charm.


symptosis

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


Willmore energy

Willmore energy (plural Willmore energies)

Named after the English geometer Thomas Willmore.

(differential geometry) A quantitative measure of how much a given surface deviates from a round sphere.


centerpoint

(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 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

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

(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


Z-shaped

Shaped like the letter "Z" or its mirror image.


X-shaped

x-förmig

shaped like the 24th letter of the alphabet


Y-shaped

An immunoglobulin is a large, Y-shaped protein.

The constellation Virgo is a Y-shaped group, of which α, the star at the foot, is the well-known Spica, a star of the first magnitude.

1920, Edward Carpenter, Pagan and Christian Creeds, New York: Harcourt, Brace and Co., published 1921:


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


exponential decay

exponential decay (countable and uncountable, plural exponential decays)

The service value of the capital stock, i.e. the product of its price and quantity component, is decomposed into a depreciation component and an interest component, similar to the decomposition which holds under exponential decay.

2017, E. Biørn, Taxation, Technology, and the User Cost of Capital, →ISBN, page 239:

A decrease where the rate of decrease is proportional to the amount remaining.


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

(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


P-shaped

Shaped like the letter "P" or its mirror image


exinscribed

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


golyhedron

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


F-shaped

Shaped like the letter "F" or its mirror image


K-shaped

Shaped like the letter "K" or its mirror image


J-shaped

Shaped like the capital letter "J" or its mirror image


Q-shaped

What I want here to carry away from the painting of balloons is a certain ontological observation. What both indicates a balloon and portrays it as, say, red is a single object – a red, Q-shaped pigment patch.

2012, J. F. Rosenberg, Linguistic Representation, page 100:

Shaped like the capital letter Q or its mirror image


H-shaped

Н-обра́зный

having a shape similar to a capital letter H


D-shaped

Shaped like the letter "D" or its mirror image


E-shaped

Shaped like the capital letter "E" or its mirror image


G-shaped

Shaped like the capital letter "G" or its mirror image


B-shaped

Shaped like the capital letter "B" or its mirror image


R-shaped

Shaped like the capital letter "R" or its mirror image


N-shaped

Shaped like the capital letter "N" or its mirror image.


O-shaped

Shaped like the capital letter "O", circular.

O-förmig

shaped like the 15th letter of the alphabet


I-shaped

工字

having a shape similar to a capital letter I with serifs

Shaped like the capital letter "I" with serifs.

Shaped like the capital letter "I" without serifs.


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.


mushroom-shaped

The deadman's pedals, which can be switched for use with either fitted or unfitted stock, are of a new type, mushroom-shaped and designed for operation with minimum fatigue.

1962 June, “The design of the S.R. electro-diesels”, in Modern Railways, page 395:

Having the characteristic shape of a mushroom (as a pillar with a broad, circular top)


rectanguloid

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


cup-shaped

Having the hollow form of a cup (but without the handle).


spheroform

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


nonhypotenuse

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


Cauchy problem

After French mathematician Augustin Louis Cauchy.

In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations.

2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41:

To emphasize the similarity between the characteristic method and the probabilistic approach we recall that the method of characteristics allows [one] to reduce the Cauchy problem for a first order PDE to a Cauchy problem for an ODE while the probabilistic approach allows [one] to reduce the Cauchy problem for a second order PDE to a Cauchy problem for an SDE (stochastic differential equation).

2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2:

In this monograph we discuss the C#x5C;infty well-posedness of the Cauchy problem for hyperbolic systems.

2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v:

partial differential equation
partial differential equation

(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.

problema di Cauchy

class of problem involving simultaneous PDEs


unode

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


golden spiral

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


logarithmic spiral

logarithmic spiral (plural logarithmic spirals)

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


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.


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:

(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.

(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.


Jordan curve theorem

There is a vast generalization of the Jordan curve theorem to higher dimensions.

1995, William Fulton, Algebraic Topology: A First Course, Springer, page 343:

The Jordan curve theorem asserts that a simple closed curve in the complex plane divides the plane into exactly two connected components, one bounded (the "inside") and the other unbounded (the "outside" ).

2001, Theodore Gamelin, Complex Analysis, Springer, page 249:

The principal reason that orbit behavior in the plane is more limited than in higher dimensions is the Jordan curve theorem. Local sections will be the primary tool used to exploit the Jordan curve theorem.

2004, Nelson G. Markley, Principles of Differential Equations, Wiley, page 193:

(topology) The theorem that states that a simple closed curve (Jordan curve) divides the plane into precisely two distinct areas.


spherical cone

(geometry) A cone with a spherical base.


transflection

transflection (countable and uncountable, plural transflections)

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

(geometry) Having opposed trapeziform faces.


symmedian

(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).


π Day

As Quakers, we do not as a matter of religion celebrate any holidays (more accurately, all days are to be considered holidays because they have equal potential for revealing the Divine to us), but as a cultural matter we celebrate all kinds: Hindu, Christian, Jewish, ethnic, birthdays of famous composers, artists, writers, and scientists, π Day (March 14th – we bake several pies and divide them into fractions before eating), even some we just make up ourselves.

1999, David H. Albert, “Exits and Entrances”, in And the Skylark Sings with Me: Adventures in Homeschooling and Community-based Education, Gabriola Island, B.C.: New Society Publishers in cooperation with Holt Associates/Growing Without Schooling, →ISBN, page 128:

Likewise, March 14 in North America has been π Day, since in the USA the month is written before the day ('314'). In schools throughout North America, it has become a reason for mathematical projects, especially focussing on π.

2012 June 19, Jonathan M[ichael] Borwein, “The Life of Pi: From Archimedes to Eniac and Beyond”, in David H[arold] Bailey, Jonathan M. Borwein, Pi: The Next Generation: A Sourcebook on the Recent History of Pi and Its Computation, Switzerland: Springer Nature, published 2016, →DOI, →ISBN, page 446:


metric tensor

2015, 1:10:00–1:11:15 from the start, in 1 Exterior Product, Fermions (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):

(differential geometry) a symmetric bilinear form which is non-degenerate (i.e., having all non-zero eigenvalues); a differential of distance on a manifold


Laplace-de Rham operator

Laplace-de Rham operator (plural Laplace-de Rham operators)

(differential geometry) A generalization of the Laplace operator; the anticommutator of the exterior derivative and the codifferential.

Laplace operators in differential geometry


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)


cevian

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


isotomic

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

affix

(geometry) Having equal but reflected cevians


hexadecagon

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


enneadecagon

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


leaf-shaped

en-adj

leaf-shaped (comparative more leaf-shaped, superlative most leaf-shaped)

more leaf-shaped (comparative)
most leaf-shaped (superlative)


icositetragon

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


hexacontagon

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


angle of depression

en-noun

angle of depression (plural angles of depression)

The angle formed by the line of sight and the horizontal plane for an object below the horizontal.


adjacent angle

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


opposite angle

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


tetradecahedron

tetradecahedron (plural tetradecahedra or tetradecahedrons)

tetradecahedra (plural)
tetradecahedrons (plural)

(geometry) A polyhedron with fourteen faces.


law of tangents

(trigonometry) A statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.


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)

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.


special function

special function (plural special functions)

Any of various specific mathematical functions that have more or less established names and notations due to their importance.


trigintaduonion

The 32-dimensional compounding fields and their quantum interplays in the trigintaduonion space can be presented by analogy with octonion and sedenion electromagnetic, gravitational, strong and weak interactions.

2007, Zihua Weng, Compounding Fields and Their Quantum Equations in the Trigintaduonion Space:

In this paper we introduce efficient algorithm for the multiplication of trigintaduonions. The direct multiplication of two trigintaduonions requires 1024 real multiplications and 992 real additions.

2014, Aleksandr Cariow, Galina Cariowa, “An algorithm for multipication of trigintaduonions”, in Journal of Theoretical and Applied Computer Science, volume 8, number 1:

The trigintaduonions form a 32-dimensional Cayley-Dickson algebra. In this paper, we introduce the k-Fibonacci and k-Lucas trigintaduonions. Moreover, we give some properties of these trigintaduonions and derive relationships between them.

2018, Kübra Gül, “On-Fibonacci and-Lucas Trigintaduonions”, in International Journal of Contemporary Mathematical Sciences, volume 13, number 1:

(mathematics) A 32-dimensional hypercomplex number that is a nonassociative extension of a sedenion.


algebraic variety

algebraic variety (plural algebraic varieties)

If #x5C;mathcal#x7B;W#x7D; 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#x5C;in#x5C;mathcal#x7B;W#x7D; axe taken to be the germs of functions on #x5C;mathcal#x7B;W#x7D; induced by functions in the stalk #x5C;mathfrak#x7B;R#x7D;#x5F;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,#x5C;mathfrak#x7B;R#x7D;), 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


ring-shaped

en-adj

ring-shaped (comparative more ring-shaped, superlative most ring-shaped)

more ring-shaped (comparative)
most ring-shaped (superlative)

Having the shape of a ring; circular; annular.


almond-shaped

almond-shaped (comparative more almond-shaped, superlative most almond-shaped)

more almond-shaped (comparative)
most almond-shaped (superlative)


top-shaped

top-shaped (comparative more top-shaped, superlative most top-shaped)

more top-shaped (comparative)
most top-shaped (superlative)


algebraic subvariety

en-noun

algebraic subvariety (plural algebraic subvarieties)

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


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.


wedge-shaped

en-adj

wedge-shaped (comparative more wedge-shaped, superlative most wedge-shaped)

more wedge-shaped (comparative)
most wedge-shaped (superlative)


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#x5C;mapstoC(R) from the category #x5C;textit#x7B;B-ALG#x7D; (also written as #x5C;textit#x7B;ALG#x7D;#x5F;#x7B;#x2F;#x5C;mathfrak#x7B;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

en-noun

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#x5C;subset#x5C;mathbb#x7B;C#x7D;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 #x5C;mathbb#x7B;C#x7D;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.