Hjelmslev's theorem

(geometry) a theorem stating that if a set of points P_i lying on a line is isometrically mapped to points P_i' on a different line, then the midpoints of the segments P_iP_i' also lie on a line.

Johannes Hjelmslev
en:Hjelmslev's theorem


arcdegree

/ˈɑːkdɪˌɡɹiː/

/ˈɑɹkdɪˌɡɹiː/

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(geometry) A unit of measurement of angle equal to ¹⁄₃₆₀ of a circle's circumference.


semicone

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algebraic analysis

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, https://books.google.com.au/books?id=qYUrDgAAQBAJ&printsec=frontcover&dq=%22algebraic+analysis%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiK5MSlvsLsAhWXbn0KHbPuBPQQ6AEwAnoECAYQAg#v=onepage&q=%22algebraic%20analysis%22&f=false.

In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^([sic]) partial differential operators.

1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,
Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

partial differential equation
partial differential equation

(algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).


equiangulation

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(geometry) Conversion to equiangular form


subsimplex

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(geometry) A polyhedron at the intersection of simplexes.


subsimplicial

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(geometry) Relating to, or composed of subsimplexes


Heesch number

(geometry) The maximum number of layers of copies of the same shape that can surround a given shape.


Heesch's problem

(geometry) The mathematical problem of determining the set of numbers that can be Heesch numbers.


QTM

en-noun

QTM (countable and uncountable, plural QTMs)

quaternary triangular mesh
quaternary triangular mesh#English

(geometry) Abbreviation of quaternary triangular mesh.


Morley's trisector theorem

Named for Anglo-American mathematician Frank Morley, who discovered it in 1899.

Among the theorems proved by Wu's method are Simon's theorem, the nine point circle theorem, Pascal's theorem, Feuerbach's theorem, Morley's trisector theorem and Thébault's conjecture which was open over forty years and was proved only in 1983.

1989, Paul GochetEric Grégoireet al., From Modal Logic to Deductive Databases, Wiley, page 8:

1996, Dongming Wang, GEOTHER: A Geometry Theorem Prover, M. A. McRobbie, J. K. Slaney (editors), Automated Deduction - Cade-13: 13th International Conference, Proceedings, Springer, LNAI 1104, page 166,
The provers are practically efficient: a number of difficult theorems including Morley's trisector theorem, Thebault's recently confirmed conjecture and MacLane's non-existence of real 83 configuration have been proved by them in the matter of seconds.

equilateral triangle
equilateral triangle

(geometry) A theorem stating that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.


algebraic K-theory

Algebraic cycles are typically studied by imposing one of several equivalence relations. The equivalence relation most relevant for algebraic K-theory is rational equivalence.

1999, E. M. Friedlander, “Lecture VII. Beilinson's vision”, in H. Bass, A. O. Kuku, C. Pedrini, editors, Algebraic K-theory And Its Applications, World Scientific, page 61:

Algebraic K-theory has two components: the classical theory which centers around the Grothendieck group K#95;0 of a category and uses explicit algebraic presentations and higher algebraic K-theory which requires topological or homological machinery to define.

2013, Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, American Mathematical Society, page ix:

2014, Daniel Scott Farley, Ivonne Johanna Ortiz, Algebraic K-theory of Crystallographic Groups, Springer, Lecture Notes in Mathematics 2113, page 1,
Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring R instead of a field. […] Algebraic K-theory plays an important part in many areas of mathematics, especially number theory, algebraic topology and algebraic geometry.

(algebraic geometry) K-theory studied from the point of view of algebra.

K-théorie algébrique

K theory studied from the point of view of algebra


Witt group

Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.

groupe de Witt

group of equivalence classes found within a field, variety or category

equivalence relation
equivalence relation

metabolic quadratic space
metabolic quadratic space

(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;

Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
(algebraic geometry, by extension) given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces
(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.


Reuleaux polygon

Reuleaux polygon (plural Reuleaux polygons)

Named after Franz Reuleaux, 19th-century German engineer.

(geometry) A curve of constant width made up of circular arcs of constant radius.


algebraic fundamental group

algebraic fundamental group (plural algebraic fundamental groups)

1997, Frans Oort, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, Cambridge University Press, London Mathematical Society, page 78,
Serre constructed an example of an algebraic variety X over a number field K plus two embeddings of K into ℂ such that the geometric fundamental groups of X_1( ℂ) and X_2( ℂ) are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).

Neves and Papadakis ([NP09]) constructed a numerical Campedelli surface with algebraic fundamental group #92;mathbbZ#95;6, while Lee and Park ([LP09]) constructed one with algebraic fundamental group #92;mathbbZ#95;2, and one with algebraic fundamental group #92;mathbbZ#95;3 was added in the second version of the same paper.

2011, Ingrid Bauer, Fabrizio Catanese, Roberto Pignatelli, “Surfaces of general type with geometric genus zero: a survey”, in Wolfgang Ebeling, Klaus Hulek, Knut Smoczyk, editors, Complex and Differential Geometry, Springer, page 11:

In order to work in arbitrary characteristic, we should use the algebraic fundamental group, which we denote by #92;pi#95;1. (A very good introduction to algebraic fundamental groups is [24].[…])

2013, János Kollár, “Rationally Connected Varieties and Connected Groups”, in Károly Böröczky, Jr., János Kollár, Szamuely Tamas, editors, Higher Dimensional Varieties and Rational Points, Springer, page 70:

(algebraic geometry) A group which is an analogue for schemes of the fundamental group for topological spaces.


anabelian geometry

anabelian geometry (countable and uncountable, plural anabelian geometries)

We shall say more about the noncommutative nature of anabelian geometries after we state the main theorem (Theorem 1.17 below) on these geometries.

1999, Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, American Mathematical Society, page 63:

The basic idea of Grothendieck's anabelian geometry is that under certain "anabelian hypotheses" the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme is functorially encoded in its étale fundamental group.
Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.

2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 519:

Therefore, the time seems right to encourage a much broader understanding of the arithmetic issues surrounding anabelian geometry and its ramifications.
While the overall importance of the theorems of anabelian geometry appears to be widely acknowledged, there is as yet not much specific knowledge within the arithmetic geometry community of its coherent body of concepts and philosophy, and of the new technology that yields actual results.

2012, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:

algebraic fundamental group
algebraic fundamental group

(mathematics, algebraic geometry, arithmetic geometry) A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).


trisection point

trisection point (plural trisection points)

The centroid of a triangle is, by definition, the point of intersection of the three medians: moreover, it is a trisection point of each of the medians.

1959 [Doubleday], William R. Gondin, Bernard Sohmer, Intermediate Algebra & Analytic Geometry Made Simple, 1967, W. H. Allen, page 147,
Find the trisection points of the line P_4P_5 in the same diagram.

1965, Paul A. White, Vector Analytic Geometry, Dickenson Publishing Company, page 73,
In triangle ABC, let E be the midpoint of BC and D the trisection point of AC nearest C.

1998, Paul J. Nahin, An Imaginary Tale: The Story of √, Princeton University Press, 2010, paperback, page 88,
And if λ=2, then P is a trisection point of P_1P_2 and
z=(z_1+2z_2)/3.
Notice that this trisection point is closer to P_2 than it is to P_1. The other trisection point, the one that is closer to P_1 than it is to P_2, occurs for λ= tfrac 1 2, which gives
z=(2z_1+z_2)/3.

(geometry) Either of two points which trisect a given line segment.


pentalateral

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(geometry) A decagon whose opposite sides are parallel.


cocked hat curve

cocked hat curve (plural cocked hat curves)

From the resemblance to the hat also called a bicorn.


plane of symmetry

plane of symmetry (plural planes of symmetry)

(geometry) A planene about which a geometric figure is symmetric (by reflection)


circumsphere

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(geometry) A sphere that is circumscribed about a polyhedron and touches each of its verticies


Euler's rotation theorem

Named after Swiss mathematician Leonhard Euler (1707–1783), who proved the theorem in 1775.

named-after

Otherwise, this approach will be valid subject to the previous rotation of the axes according to Euler's Rotation Theorem.

2003, On the Move to Meaningful Internet Systems 2003: CoopIS, DOA, and ODBASE, Proceedings, page 136:

2005, JSME International Journal: Mechanical systems, machine elements and manufacturing, Volume 48, Issue 4, Japan Society of Mechanical Engineers, page 501,
According to Euler's rotation theorem, any finite rotation transformations may be described using only three angles defined as Euler angles.

From Euler's rotation theorem we know that an arbitrary rotation in three dimensions can be parameterized by three independent variables called the Euler angles, each rotating around an orthogonal axis.

2006, Patricia J. Lee, Quantum Information Processing with Two Trapped Cadmium Ions, University of Michigan, page 40:

(geometry) A theorem which states that, in 3-dimensional space, any displacement of a rigid body such that some point on it remains fixed is equivalent to a single rotation about some axis that runs through said point.


cubicuboctahedron

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(geometry) Either of two star polyhedra whose square faces are parallel to a square and whose triangular faces are parallel to those of an octahedron


isohedron

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star polyhedron

(geometry) A polyhedron that is partially nonconvex.


small cubicuboctahedron

en-noun

small cubicuboctahedron (plural small cubicuboctahedra)

(geometry) A uniform star polyhedron that has faces of eight triangles, six squares and six octagrams


great cubicuboctahedron

great cubicuboctahedron (plural great cubicuboctahedra)

(geometry) A nonconvex uniform polyhedron that has faces of eight triangles, six squares and six octagrams


space-filling

(geometry) That fills a closed (typically 2-dimensional or 3-dimensional) space.


Euler angle

Named after Swiss mathematician Leonhard Euler (1707–1783).

Euler angles can also represent the orientation of a mobile frame of reference in physics or the orientation of a basis in 3-dimensional linear algebra.

The use of Euler angles or Quaternions to represent entity (i.e., vehicle) orientation in the draft military standard for Distributed Interactive Simulation has drawn fierce debate.

1991, “AIAA Flight Simulation Technologies Conference: A Collection of Technical Papers”, in AIAA, page 271:

We have not drawn a picture, since Euler angle pictures of three-dimensional rotations are notoriously difficult to comprehend; what is very useful is for the reader to make a set of wire axes with labels to help follow the sequence of rotations.

2006, Don Koks, Explorations in Mathematical Physics: The Concepts Behind an Elegant Language, Springer, page 171:

Why do we need to know what Euler angles are? Because we use them. Many commercial 3D computer animation packages that are available today use Euler angles to describe and change orientations of objects.

2008, Midori Kitagawa, Brian Windsor, MoCap for Artists: Workflow and Techniques for Motion Capture, Elsevier (Focal Press), page 188:

(geometry) Any one of three angles which together describe the orientation of a rigid body with respect to a fixed three-dimensional coordinate system, according to a formalism introduced by Leonhard Euler.

eulerscher Winkel

any one of three angles that describe orientation in a 3-d coordinate system


costraight

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Thales's theorem

Generally attributed to Thales of Miletus.

(geometry) A theorem stating that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle.

intercept theorem
intercept theorem#English


quarter circle

(geometry) A quarter of a circle. The shape formed from an arc that is a quarter of the circumference of a circle.


polyhedrometry

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(geometry) The measurement of polyhedra.


hexangle

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quindecangle

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(obsolete, geometry) Synonym of quindecagon.


quadriquadric

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(geometry, obsolete) A curve that is the complete intersection of two quadric surfaces.


Boothian

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(geometry) Of or relating to a tangential coordinate system invented by James Booth.


metageometrical

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pentagonohedron

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(geometry) Any polyhedron that has pentagonal faces.


isoperimeter

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(geometry) A figure that has the same perimeter as another


homocentrical

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(geometry) Alternative form of homocentric (“having the same centre”)


velary

(geometry, obsolete) Relating to a curve representing the motion of a sail blown by the wind.


Sumihiro's theorem

(algebraic geometry) A theorem stating that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.


Białynicki-Birula decomposition

Białynicki-Birula decomposition (plural Białynicki-Birula decompositions)

The description depends on a partial ordering of the fixed point components which arises from the Białynicki-Birula decompositions of X.

1984, Pacific Journal of Mathematics, volumes 114-115, page 149:

Section 2 describes a C*-action on Grassmann manifolds and gives the corresponding Białynicki-Birula decomposition.

1988, Mathematica Scandinavica, volume 62, page 5:

(algebraic geometry) A T-stable cellular decomposition of a smooth T-variety.


Schönhardt polyhedron

Schönhardt polyhedron (plural Schönhardt polyhedra)

Named after German mathematician Erich Schönhardt, who described it in 1928.

(geometry) The simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices.


octakishexahedron

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(geometry, rare) A polyhedron with 48 faces.


runcicantellation

(geometry) The simultaneous state of runcination and cantellation.


omnitruncation

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Wythoff construction
Wythoff construction

(geometry) An operation applied to a regular polytope in a Wythoff construction that creates a maximum number of facets.


Wythoff construction

Wythoff construction (plural Wythoff constructions)

Named after the Dutch mathematician Willem Abraham Wythoff (1865–1939).

(geometry) A method for constructing a uniform polyhedron or plane tiling.

Willem Abraham Wythoff
Wythoff construction


Ptolemy's theorem

Named after the Greek astronomer and mathematician Ptolemy.

cyclic quadrilateral
cyclic quadrilateral

(geometry) A theorem that states in a cyclic quadrilateral, the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the two pairs of opposite sides.


didodecahedron

didodecahedron (plural didodecahedra or didodecahedrons)

didodecahedra (plural)
didodecahedrons (plural)

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Didodecahedrons are never seen in crystals, as predominating forms; but they are frequently met with as modifying faces, in combinations, principally in those in which the 6-sided prism predominates.

1853, “Crystallography”, in Th[omas] Forrest Betton, transl., edited by James C[urtis] Booth and William L. Faber, Elements of Chemistry. For the Use of Colleges, Academies, and Schools., 2nd edition, volume I, Philadelphia, Pa.: Clark & Hesser, translation of original by [Henri] V[ictor] Regnault, section “The six systems of crystalline forms”, subsection 3 (Hexagonal system), page 34:

[…]a pair of opposite edges of every face are parallel, so that the form shows prominent zonal relationships which are not characteristic of didodecahedra in general.

1946, F[rank] C[oles] Phillips, “The Thirty-two Classes”, in An Introduction to Crystallography, Longmans, Green and Co., part I (The External Symmetry of Crystals), pages 145–146:

(geometry) A solid with 24 faces consisting of scalene triangles.


mucube

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(geometry) A regular skew apeirohedron with six squares around each vertex, formed by an infinite number of cubic cells (specifically, cubes missing two opposite faces each, around cubic empty spaces with no faces) in Euclidean 3D space.


mutetrahedron

mutetrahedron (plural mutetrahedra or mutetrahedrons)

mutetrahedra (plural)
mutetrahedrons (plural)

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truncated tetrahedron
truncated tetrahedron

(geometry) A regular skew apeirohedron with six hexagons around each vertex, formed by an infinite number of truncated tetrahedron-like cells (specifically, truncated tetrahedra missing their triangle faces, with the resulting triangle-shaped holes joined to form empty spaces in the shape of faceless tetrahedra) in Euclidean 3D space.

六角六片三角孔ねじれ正多面体

ろっかくろくへんさんかくこうねじれせいためんたい


muoctahedron

muoctahedron (plural muoctahedra or muoctahedrons)

muoctahedra (plural)
muoctahedrons (plural)

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truncated octahedra
truncated octahedron

(geometry) A regular skew apeirohedron with four hexagons around each vertex, formed by an infinite number of octahedron-like cells (specifically, truncated octahedra missing their square faces, and joined by the resulting square-shaped holes) in Euclidean 3D space.

六角四片四角孔ねじれ正多面体

ろっかくしへんしかくこうねじれせいためんたい


Donaldson-Thomas theory

Donaldson-Thomas invariant
Donaldson-Thomas invariant

(algebraic geometry) The theory of Donaldson-Thomas invariants.


Donaldson-Thomas invariant

Donaldson-Thomas invariant (plural Donaldson-Thomas invariants)

Introduced in 1998 by Simon Donaldson and Richard Thomas.

Calabi-Yau threefold
Calabi-Yau manifold#English

(algebraic geometry) Given a compact moduli space of sheaves on a Calabi-Yau threefold, its Donaldson-Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class.


Calabi-Yau manifold

Calabi-Yau manifold (plural Calabi-Yau manifolds)

The name was coined by Candelas et al. (1985), after Eugenio Calabi (1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978), who proved the Calabi conjecture.

(algebraic geometry) A kind of manifold having properties, such as Ricci flatness, that yield applications in theoretical physics.


obliquangular

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an obliquangular triangle

(archaic, geometry) Formed of oblique angles.


lateral surface

lateral surface (plural lateral surfaces)

(geometry) All of the sides of the object, excluding its base and top (when they exist).

سطح جانبي

all of the sides of the object, excluding its base and top (when they exist)


multangle

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inellipse

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(geometry) An ellipse inscribed in a triangle.


pursuit curve

(geometry) A geometrical model which simulates the path taken by a pursuer in the course of its pursuit.


octiamond

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(geometry) A polyiamond made up of eight triangles.


cylindroidal

en-adj

cylindroidal (comparative more cylindroidal, superlative most cylindroidal)

more cylindroidal (comparative)
most cylindroidal (superlative)

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(geometry) Having the form of a cylindroid.


grand 600-cell

(geometry) A regular star 4-polytope with Schläfli symbol {3,3,5/2}


great icosahedron

en-noun

great icosahedron (plural great icosahedra or great icosahedrons)

great icosahedra (plural)
great icosahedrons (plural)

Kepler-Poinsot polyhedra
Kepler-Poinsot polyhedra

(geometry) One of four Kepler-Poinsot polyhedra with Schläfli symbol {3,5⁄2}.


hyperbolic dodecahedron

hyperbolic dodecahedron (plural hyperbolic dodecahedrons)

(hyperbolic geometry) A hyperbolic version of the Euclidean dodecahedron.


great dodecahedron

great dodecahedron (plural great dodecahedra or great dodecahedrons)

great dodecahedra (plural)
great dodecahedrons (plural)

(geometry) A Kepler solid, with the Schläfli symbol {5,5/2}.


Cavalieri's principle

method of indivisibles
method of indivisibles

(geometry) A development of the method of indivisibles, dealing with two-dimensional area or three-dimensional volume based on the intersections of lines or planes surrounding a region. It was an early step toward integral calculus.


wallpaper group

wallpaper group (plural wallpaper groups)

(geometry) A mathematical group that classifies two-dimensional repeating patterns based on their symmetries.


hexadecahedroid

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(geometry) The 16-cell or hexadecachoron.


bispherical

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Varignon's theorem

Named after French mathematician Pierre Varignon.

Varignon parallelogram
Varignon parallelogram

(geometry) A statement in Euclidean geometry that deals with the construction of a particular parallelogram (the Varignon parallelogram) from an arbitrary quadrilateral.


planigon

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(geometry) A convex polygon that can fill the plane with only copies of itself.


nonintersector

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(geometry) That which is not an intersector.


square degree

(geometry) A unit of solid angle equal to π²/32400 steradian


tricuspal

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(geometry, dentistry) Consisting of, or relating to, three cusps.


kissing number

(geometry) For a given space, the maximum number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere.


Pappian

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a Pappian plane

Pappus's hexagon theorem
Pappus's hexagon theorem

(geometry) For which Pappus's hexagon theorem is valid.


icositrigon

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(geometry) A polygon with 23 sides and 23 angles.


hyperprism

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(geometry) A prism that extends into hyperspace, having more than three dimensions.


hypersolid

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(geometry) A solid that extends into hyperspace, having more than three dimensions.


hypertetrahedron

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(geometry) An analog of a tetrahedron that extends into hyperspace, having more than three dimensions.


oblique pyramid

oblique pyramid (plural oblique pyramids)

(geometry) A nonright pyramid, where the apex is not centered over the base.


oblique reflection

oblique reflection (plural oblique reflections)

(geometry) In Euclidean geometry, an ordinary reflection not requiring that the reflection be performed using perpendiculars.


Keller graph

(geometry) Any of a certain family of graphs related to Keller's conjecture, such that the vertices of the Keller graph of dimension n are the 4ⁿ elements (m₁,...,mₙ) where each m is 0, 1, 2, or 3, and two vertices are joined by an edge if they differ in at least two coordinates and differ by exactly two in at least one coordinate.


Keller's conjecture

The conjecture was introduced by Ott-Heinrich Keller (1930).

(geometry) The conjecture that, in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other.


focaloid

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(geometry) A shell bounded by two concentric, confocal ellipses (in two dimensions) or ellipsoids (in three dimensions).


parallel transport

parallel transport (countable and uncountable, plural parallel transports)

(geometry) A way of transporting geometrical data along smooth curves in a manifold.


Star of Lakshmi

"Lakshmi" from the Hindu god Lakshmi and the "star" word in the noun denotes the octagram.

(geometry) A special octagram, a regular compound polygon, represented by Schläfli symbol {8/2} or 2{4}, made from two congruent squares with the same center at 45° angles.


Boerdijk-Coxeter helix

Named after H. S. M. Coxeter and A. H. Boerdijk.

(geometry) A linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices.


Weaire-Phelan structure

Weaire-Phelan structure (plural Weaire-Phelan structures)

Named after Denis Weaire and Robert Phelan, who in 1993 found that this structure was a better solution to the Kelvin problem of tiling space by equal-volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.

(geometry) A three-dimensional structure representing an idealized foam of equal-sized bubbles, with two different shapes.


Tschirnhausen cubic

Tschirnhausen cubic (plural Tschirnhausen cubics)

Named after Ehrenfried Walther von Tschirnhaus (1651–1708), German mathematician, physicist, physician, and philosopher. The name was first used in a 1900 paper by R. C. Archibald.

(geometry) A plane curve defined, in its left-opening form, by the polar equation r=a sec ³(θ/3) where sec is the secant function.


hexahectaenneacontakaiheptagon

Analysable as hexahecta- (“six hundred”) (from hexa- + hect(a)-) + enneaconta- (“ninety”) (from ennea- + the common element in Ancient Greek ἑξήκοντα (hexḗkonta, “sixty”), τεσσαράκοντα (tessarákonta, “forty”), etc.) + Ancient Greek καί (kaí, “and”) + hepta- (“seven”) + -gon.
Coined humorously to describe the shape of the US state of Colorado.

der

Colorado is actually a hexahectaenneacontakaiheptagon, meaning it has 697 sides, according to a 2018 bigthink.com article and Kutztown University's math whizzes who create names for polygons.

2019 October 16, James Burky, “Rectangle state? Nope. Colorado has 697 sides, not four.”, in The Denver Post, archived from the original on 2023-01-08:

A study in 2018 suggested that as a result of the challenges of surveying in the nineteenth century, Colorado has no fewer than 697 sides, technically making it a "hexahectaenneacontakaiheptagon."

2020, Derek Everett, Colorado Day by Day, Boulder, CO: University Press of Colorado, →ISBN, page 212:

The state of Colorado may look like a rectangle (four straight sides, four right angles) but it actually has 697 sides, making it a hexahectaenneacontakaiheptagon.

2020, Bathroom Readers' Institute, Uncle John's Greatest Know on Earth Bathroom Reader, San Diego, CA: Portable Press, →ISBN:


semidisk

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kaleidocycle

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(geometry) A flexible polyhedron connecting six tetrahedra (or disphenoids) on opposite edges into a cycle.


Lemoine point

The French mathematician Émile Lemoine proved the existence of such points in 1873.

(geometry) The intersection of the three symmedians of a triangle.


spherical triangle

en-noun

spherical triangle (plural spherical triangles)

If the three sides of a spherical triangle be each of them equal to a quadrant, the polar triangle will coincide with it; for each of the angular points will be the pole of the side opposite to it.

1830, Pierce Morton, Geometry, Plane, Solid, and Spherical, in Six Books, Baldwin and Craddock, page 189:

570. Theorem. The sum of the sides of a spherical triangle may be anything between 0 and a circumference.

1876, Edward Olney, A Treatise on Special Or Elementary Geometry, Sheldon & Company, page 220:

1893, Crossley William Crosby Barlow, George Hartley Bryan, Elementary Mathematical Astronomy, W. B. Clive, page v,
A spherical triangle, like a plane triangle, has six parts, viz., its three sides and its three angles. The sides are generally measured by the angles they subtend, so that the six parts are all expressed as angles.
Any three parts suffice to determine a spherical triangle, but there are certain "ambiguous cases" when the problem admits of more than one solution. The formulæ required in solving spherical triangles form the subject of Spherical Trigonometry, and are in every case different from the analogous formulæ in Plane Trigonometry. There is this further difference, that a spherical triangle is completely determined if its three angles are given.

The angles in a spherical triangle do not have to add up to 180 degrees. It is possible for a spherical triangle to have 3 right angles.

2012, Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, 32nd edition, Taylor & Francis (CRC Press / Chapman & Hall), page 218:

(geometry, spherical geometry) A triangle, described on the surface of the sphere, whose each side is an arc of some great circle.

Kugeldreieck

Triangle on the surface of a sphere, whose sides are arcs of great circles


subchord

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nonant

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Borrowed from Latin nōnum, accusative singular of nōnus (“ninth part of something, ninth”).

(geometry) One ninth of a circle or disc; a sector with an angle of 40°.


(geometry) A cube in four dimensions or four-dimensional space; a tesseract, 8-cell or octachoron; made of eight cubical cells or eight cubes.


scalenon

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(geometry, obsolete) A scalene triangle.


pentille

(geometry) A tessellation of the plane consisting of irregular pentagons.


monohedrally

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The extent of our ignorance is further illustrated by the fact that we do not know all the different shapes of convex pentagons which will tile the plane monohedrally.

1987, Branko Grünbaum, G[eoffrey] C[olin] Shephard, Tilings and Patterns, New York, N.Y.: W. H. Freeman and Company, →ISBN, page 22:

(geometry) In a monohedral manner; involving only copies of the original tile (allowing for translation, rotation, and reflection).


monotile

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In the following section we will describe families of tilings by a curious three-dimensional aperiodic monotile whose matching rule — for some values of its parameter — is weak but not strong.

1995, Marjorie Senechal, Quasicrystals and Geometry, Cambridge, Cambridgeshire: Cambridge University Press, →ISBN, page 208:

And I am sorry I never delivered on the four-dimensional Poultry design. It turns out John Horton Conway found four-dimensional and five-dimensional aperiodic monotiles sixty years ago, but it's not too well documented.

1997, Rudy Rucker, Freeware (Ware Tetralogy; 3), New York, N.Y.: Avon Books, →ISBN, page 69:

This naturally led to serious rumination about the possible existence of a single tile, or monotile, that forces non-periodic global structure.

2007 March, Joshua E. S. Socolar, “More ways to tile with only one shape polygon”, in The Mathematical Intelligencer, volume 29, number 2, New York, N.Y.: Springer, →DOI, →ISSN, →OCLC, page 33:

An aperiodic monotile never repeats a formation, no matter how long the pattern.

2023 April 4, Matthew Cantor, “'The miracle that disrupts order': mathematicians invent new 'einstein' shape”, in The Guardian, London: Guardian News & Media, →ISSN, →OCLC, archived from the original on 2023-04-04:

(geometry) A prototile tiling monohedrally; any shape able to completely tile some space on its own (allowing for translation, rotation, and reflection).


duocylinder

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(geometry) A geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks, and analogous to a cylinder in 3-space.


ex-tangential

(geometry) Pertaining to external tangents.


direction vector

direction vector (plural direction vectors)

Coordinate term: position vector

(geometry) A unit vector used to represent spatial direction, commonly denoted as d.


Gauss-Codazzi equation

Gauss-Codazzi equation (plural Gauss-Codazzi equations)

Named after Carl Friedrich Gauss and Delfino Codazzi.

pseudo-Riemannian manifold
pseudo-Riemannian manifold

(geometry) Any of a group of fundamental formulas in Riemannian and pseudo-Riemannian geometry that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.


Hjelmslev transformation

Hjelmslev transformation (plural Hjelmslev transformations)

(geometry) A method for mapping an entire hyperbolic plane into a circle with a finite radius.


Petersen-Morley theorem

Named after Johannes Trolle Petersen (later Johannes Hjelmslev) and Frank Morley.

(geometry) A theorem stating that, if a, b, c are three general skew lines in space, if a′, b′, c′ are the lines of shortest distance respectively for the pairs (b,c), (c,a) and (a,b), and if p, q and r are the lines of shortest distance respectively for the pairs (a,a′), (b,b′) and (c,c′), then there is a single line meeting at right angles all of p, q, and r.


hyperboloid of revolution

hyperboloid of revolution (plural hyperboloids of revolution)

(geometry) The surface generated by rotating a hyperbola around one of its principal axes.


Cartesian equation

Cartesian equation (plural Cartesian equations)

(geometry, algebra) An equation that describes a line or plane in terms of the relationship between the coordinates on the axes involved.


Minkowski-Bouligand dimension

Minkowski-Bouligand dimension (plural Minkowski-Bouligand dimensions)

Named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.

(geometry) A way of determining the fractal dimension of a set in a Euclidean space, or more generally in a metric space. It may be visualized by imagining the fractal lying on an evenly-spaced grid and counting how many boxes are required to cover the set. The dimension is then calculated by seeing how this number changes as the grid is made finer.


glide reflection

glide reflection (countable and uncountable, plural glide reflections)

Thus every function of the form eⁱᵃz + c is the composition of a translation with a reflection and so it is a glide reflection.

1993, Saul Stahl, The Poincaré Half-plane: A Gateway to Modern Geometry, Jones & Bartlett Learning, →ISBN, page 136:

That turns out to have one small spaceship, moving diagonally up and left at c/14 with period 14 (glide reflection at generation 7):

1995 November 12, Rich Holmes, “Totalistic spaceships”, in comp.theory.cell-automata (Usenet):

Footprints along a line have glide reflection symmetry.

2004, Dorothy Koster Washburn, Donald W. Crowe, Donald Warren Crowe, Symmetry Comes of Age: The Role of Pattern in Culture, University of Washington Press, →ISBN, page 6:

(geometry) A transformation consisting of a reflection followed by a translation along the same line.


Desargues's theorem

(geometry) A theorem stating that two triangles are in perspective axially if and only if they are in perspective centrally.


R-pentomino

(geometry) A particular five-square polyomino.

I came up with an implementation of an algorithm on my Sparcstation 1 that does all 1103 generations of the R-pentomino, including the gliders, in 1.3 seconds (no display), and runs o.....o. ..o...o. ..o..o.o .o.o.... (a pattern by Wainright recommended to me by Gosper) out to 10,000 generations in 45 seconds (again, without display).

1990 January 15, Paul Dietz, “Gosper's Algorithm for Life”, in comp.theory.cell-automata (Usenet):

I've finisher debugging the code, and it works -- that is to say, its final configuration for the R-pentomino is the same as the one in my books. (It's my guess that the R-p will find any bug in any life program, no matter how obscure.)

1990 June 8, Robert Herrell, “Life without bounds”, in comp.theory.cell-automata (Usenet):

This explosive collision is "perturbed" by the permanent parts of the reflector so that at generation 242, the destroyed block is restored, a glider is released in the reverse direction, an r-pentomino is produced some distance from the stage, and a beehive is left in an awkward position close to the block.

1996 November 19, Paul Callahan, “Glider reflector in Life”, in comp.theory.cell-automata (Usenet):

(cellular automata) The methuselah based on this polyomino that, in Conway's Game of Life, has a lifespan of 1,103 generations.


paraboloid of revolution

paraboloid of revolution (plural paraboloids of revolution)

surface of revolution
surface of revolution

(geometry) A surface of revolution formed by revolving a parabola around its axis.


semiminor axis

Coordinate term: semimajor axis

(geometry) Half of the minor axis of a circle or ellipse.

demi-petit axe

half of the major axis of a circle or ellipse


Mojette transform

Mojette transform (plural Mojette transforms)

(geometry) A discrete and exact version of the Radon transform, using only additions and subtractions, and having applications in medical tomography, cryptography, etc.


lune of Hippocrates

lune of Hippocrates (plural lunes of Hippocrates)

(geometry) A lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle.


midpoint theorem

A theorem stating that if the midpoints of two sides of a triangle are connected, the resulting line segment will be parallel to the third side and have half of its length.

A theorem stating that the midpoints of parallel chords in a conic are located on a common line.


hectohedron

hectohedron (plural hectohedra or hectohedrons)

hectohedra (plural)
hectohedrons (plural)

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For quotations using this term, see Citations:hectohedron.

(rare, geometry) A polyhedron with one hundred faces.


Moser's worm problem

Formulated by the Austrian-Canadian mathematician Leo Moser in 1966.

(geometry) An unsolved problem asking for the region of smallest area that can accommodate every plane curve of length 1, i.e. the curve can be rotated and translated to fit inside the region (which in some variations of the problem is restricted to be convex).


intercept theorem

(geometry) An important elementary theorem about the ratios of various line segments that are created if two rays with a common starting-point are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles.


Brocard circle

Named after Henri Brocard (1845–1922), French mathematician.

(geometry) A circle that passes through the circumcenter and symmedian of a given triangle, and is centered at the midpoint of the line segment joining them.


Japanese theorem

A theorem stating that, no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant. Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic.

A theorem stating that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.


Varignon parallelogram

Varignon parallelogram (plural Varignon parallelograms)

(geometry) The parallelogram involved in Varignon's theorem, whose corners are the midpoints of the sides of an arbitrary quadrilateral.


Simson line

Named after Robert Simson, though it had been published prior to his work.

(geometry) Given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P.


Wallace line

Named after William Wallace, who published the concept in 1799.


folium of Descartes

(geometry) An algebraic curve defined by the implicit equation x³+y³-3axy=0.


pyramidoidal

en-adj

pyramidoidal (comparative more pyramidoidal, superlative most pyramidoidal)

more pyramidoidal (comparative)
most pyramidoidal (superlative)

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Moufang plane

(geometry) A type of projective plane whose every line is a translation line.


Pappus's hexagon theorem

Attributed to the Ancient Greek mathematician Pappus of Alexandria (c. 290–c. 350 AD).

(geometry) A theorem valid for projective planes over any field, stating that, given one set of collinear points A,B,C, and another set of collinear points a,b,c,, the intersection points X,Y,Z of line pairs Ab and aB,Ac and aC,Bc and bC are collinear, lying on the "Pappus line". These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc.


myriahedron

en-noun

myriahedron (plural myriahedrons or myriahedra)

myriahedrons (plural)
myriahedra (plural)

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For quotations using this term, see Citations:myriahedron.

(geometry) A polyhedron with 10,000 faces.


kisrhombille

(geometry, rare) A uniform tiling of rhombic faces divided by central points into four triangles.


undecahedron

undecahedron (plural undecahedrons or undecahedra)

undecahedrons (plural)
undecahedra (plural)

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(geometry) A polyhedron with eleven faces.


pentacontahedron

pentacontahedron (plural pentacontahedrons or pentacontahedra)

pentacontahedrons (plural)
pentacontahedra (plural)

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From Ancient Greek πεντήκοντα (pentḗkonta) + -hedron.

suffix

(geometry) A polyhedron with fifty faces.


monohedron

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This is a reflection of the fact that the monohedron collapses asymptotically to a line (along the w₁-axis) when T -> Tₜ (t -> 0) (Widom, 1975c).

1972, Cyril Domb, M. S. Green, Joel Louis Lebowitz, Phase Transitions and Critical Phenomena - Volume 9, page 112:

(geometry) A solid figure with a single continuous surface (face) and no edges, such as a torus or spheroid.


silver ratio

(geometry) The irrational number (approximately 2.414), which is equal to the sum of its own reciprocal and two. It equals one plus the square root of two (1 + √).


duocylindrical

en-adj

duocylindrical (comparative more duocylindrical, superlative most duocylindrical)

more duocylindrical (comparative)
most duocylindrical (superlative)

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(geometry) Of or pertaining to a 4-dimensional object having the shape of a duocylinder.


faithfully flat

(algebraic geometry, of a morphism of schemes) Flat and surjective.


vanishing locus

(algebraic geometry) Synonym of zero locus.


zero locus

(algebraic geometry, of a polynomial, or, more generally, a function) The set of points (usually in affine or projective space, but more generally in the domain of the function) where the given polynomial (or function) takes the value zero.


affine scheme

locally ringed space
locally ringed space

(algebraic geometry) An abstract, geometric object in mathematics which generalizes the notion of an affine variety qua domain of a commutative ring of polynomial functions: a space which is the result of a particular construction beginning with an arbitrary commutative ring whereby elements of the ring can be thought of as functions on the space; a space which can be constructed in such a way. Formally: the spectrum of a commutative ring; a locally ringed space isomorphic to the spectrum of a commutative ring.


distinguished open set

en-noun

distinguished open set (plural distinguished open sets)

(algebraic geometry) A particularly basic kind of set, generalizing the notion of the compliment of a hypersurface, originating in the study of algebraic varieties but in modern mathematics also extending to the setting of affine schemes. Formally: (in the context of affine or projective varieties) the compliment of the zero locus of a polynomial in affine or projective space; (in scheme theory) the subset of prime ideals of a commutative ring which do not contain some element of the ring.


structure sheaf

structure sheaf (plural structure sheaves)

(sheaf theory, algebraic geometry) The sheaf associated to a ringed space.


B-sheaf

(sheaf theory, algebraic geometry) A kind of partial, simpler to describe sheaf which can be extended into a sheaf (and which is, in a technical sense, equivalent to that sheaf). For a formal definition, see B-sheaf on Wikipedia.Wikipedia.


quasi-projective

From quasi- + projective, the sense being that projective varieties are closed subsets of projective space while quasi-projective varieties are only required to be locally closed, a strictly weaker condition. The second, scheme-theoretic sense generalizes the first.

(algebraic geometry, of a variety) Locally closed with respect to the Zariski topology on projective n-space; isomorphic to such a set.

(algebraic geometry, of a scheme) Comprising an open subscheme of a projective scheme


trisectible

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(geometry) Able to be divided into three equal angles using a compass and straightedge.


trisectability

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(geometry) Ability to be divided into three equal angles using a compass and straightedge.


pentasectible

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(geometry) Able to be divided into five equal angles using a compass and straightedge.


obtusangled

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12 An obliquangled triangle is either Obtuſangled or Acutangled.
The diviſion of an obliquangled triangle is taken from the ſpeciall differences of an oblique angle. For at the 15 e iij, we were taught that an oblique angle was either obtuſe or acute: Therefore an obliquangled triangle is an obtuſeangle, and an Acutangle.

1636, Peter Ramvs [i.e., Petrus Ramus], “Of Geometry, the ſeventh Booke, Of the compariſon of Triangles”, in William Bedvvell [i.e., William Bedwell], transl., Via Regia ad Geometriam: The VVay to Geometry, […] Thomas Cotes, […] Michael Sparke, page 111:

(geometry) Obsolete form of obtuse-angled.


acutangled

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(geometry) Obsolete form of acute-angled.


obliquangled

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(geometry) Obsolete form of oblique-angled.


obliquangle

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But if it be an obliquangle triangle, to note it with the letters BCD.

1618, John Napier, translated by Edward Wright, A description of the admirable table of logarithmes, London: […] Simon Waterson, page 3:

Notwithſtanding in obliquangle triangles, although the ſhankes are not the heights, the caulſe of the truth hereof is the ſame.

1636, Peter Ramvs [i.e., Petrus Ramus], “Of Geometry, the ſeventh Booke, Of the compariſon of Triangles”, in William Bedvvell [i.e., William Bedwell], transl., Via Regia ad Geometriam: The VVay to Geometry, […] Thomas Cotes, […] Michael Sparke, page 100:

IN an obliquangle Triangle, when the termes propounded are two ſides and one Angle , or two Angles and one ſide, and yet the terme required undiſcoverable by the two laſt premiſed Problemes, you are to convert ſuch a Triangle into two Rectangle Triangles, […]

1645, Edmund Wingate, “The uſe of the Rule of Proportion in Geometrie” (chapter 5), in Edmund Wingate, transl., The uſe of the Rule of Proportion: in Arithmetique and Geometrie, London: […] M. F. for P. Stephenss, Of Sphericall Obliquangle Triangles, page 59:

And here we may take in by the way, another kind of tetragonall Plane, very variable in reſpect of it's angles, but regular-like, (as I may term it) being equilaterall, though notequi-angular , called Rhombus, being the onely obliquangle equilaterall Parallelogram, and which therefore is a Quadrat, […]

1650, John Wyberd, Tactometria. Seu, Tetagmenometria. Or the Geometry of Regulars practically proposed; […] , […] Robert Leybourn, for Nathaniel Brooks, pages 102-103:

(geometry) Obsolete form of oblique angle.


(geometry) Synonym of decagon: A polygon with ten sides and ten angles.


(geometry) Synonym of enneadecagon: A polygon with nineteen sides and nineteen angles.


(geometry, rare) Synonym of dihectagon: A polygon with two hundred sides and two hundred angles.


(geometry) Synonym of dodecagon: A polygon with twelve edges and twelve angles.


(geometry) Synonym of enneacontagon: A polygon with 90 sides and 90 angles.


(geometry) Synonym of heptagon: A polygon with seven sides and seven angles.


(geometry) Synonym of enneacontahexagon: A polygon with ninety six sides and ninety six angles.


(geometry, rare) Synonym of chiliagon: A polygon with a thousand vertices and a thousand edges.


(geometry) Synonym of hendecagon: A polygon with eleven edges and eleven angles.


(geometry) Synonym of quadrilateral: A polygon with four sides.


(geometry) Synonym of hexadecagon: A polygon with sixteen sides and sixteen angles.


(geometry) Synonym of icosihenagon: A polygon with twenty-one edges and twenty-one angles.


(geometry) Synonym of heptadecagon: A polygon with seventeen sides and seventeen angles.


(geometry) Synonym of pentagon: A polygon with five sides and five angles.


(geometry) Synonym of megagon: A polygon with one million sides.


(geometry) Synonym of icosidigon: A polygon with twenty two sides and twenty two angles.


(geometry) Synonym of hexacontagon: A 60-sided polygon.


(geometry) Synonym of pentadecagon: A polygon with fifteen sides.


(geometry, also attributively) Synonym of octagon: A polygon with eight sides and eight angles.


(geometry) Synonym of tetracontadigon: A polygon with forty two sides and forty two angles.


(geometry) Synonym of myriagon: A polygon with 10,000 sides.


(geometry) Synonym of icositetragon: A polygon with 24 sides and 24 angles.


(geometry) Synonym of triacontagon: A polygon with thirty sides.


(geometry) Synonym of tetracontakaihexagon: A polygon with forty six sides and forty six angles.


(geometry) Synonym of icositrigon: A polygon with 23 sides and 23 angles.


(geometry) Synonym of octacontagon: A polygon with 80 sides and 80 angles.


(geometry) Synonym of tetradecagon: A polygon having fourteen sides and fourteen angles.


(geometry) Synonym of tetracontagon: A 40-sided polygon.


(geometry, rare) Synonym of octadecagon: A polygon with 18 sides and 18 vertices.


(geometry) Synonym of triacontadigon: A polygon with thirty-two sides and thirty-two angles.


(geometry) Synonym of triacontatetragon: A polygon with thirty four sides and thirty four angles.


(geometry) Synonym of hexagon: A polygon with six sides and six angles.


(geometry) Synonym of heptacontagon: A polygon with seventy edges.


(geometry) Synonym of hectogon: A polygon having a hundred sides.


(geometry) Synonym of icosagon: A polygon with twenty sides and twenty angles.


(geometry) Synonym of tridecagon: A polygon having thirteen sides and thirteen angles.


(geometry) Synonym of enneagon: A polygon with nine sides; a nonagon.


harmonic division

harmonic division (plural harmonic divisions)

(mathematics, geometry) The configuration of four ordered collinear points such that their cross-ratio is -1.


improper rotation

improper rotation (plural improper rotations)

(geometry) An isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis.


Schiffler's theorem

(geometry) A theorem stating that, given a triangle △ABC with the incenter I, the Euler lines of the four triangles △BCI, △CAI, △ABI, △ABC all meet at a single point (the Schiffler point).


Schiffler point

Schiffler point (plural Schiffler points)

First defined and investigated by Schiffler et al. (1985).

(geometry) A certain type of triangle center. A triangle △ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines of the four triangles △BCI, △CAI, △ABI, △ABC.


Bevan point

Named after British engineer Benjamin Bevan.

(geometry) A triangle center, defined as the center of the Bevan circle.


Bevan circle

(geometry) The circle through the centers of the three excircles of a triangle.


Feuerbach point

Feuerbach point (plural Feuerbach points)

(geometry) A kind of triangle center, defined as the point at which the incircle and nine-point circle of the triangle are internally tangent to each other.


Clawson point

Named after John Wentworth Clawson, who published it 1925.

(geometry) A point in a triangle defined by the trilinear coordinates tan α : tan β : tan γ, where α, β, γ are the interior angles at the triangle vertices A, B, C.


Fermat-Torricelli point

Fermat-Torricelli point (plural Fermat-Torricelli points)


Torricelli point

Torricelli point (plural Torricelli points)


Fermat point

(geometry) A point such that the sum of the three distances from each of the three vertices of a given triangle to that point is the smallest possible; equivalently, the geometric median of the three vertices.


Spieker center

(geometry) The center of a Spieker circle, which is the incenter of the medial triangle.


Nagel point

Named after Christian Heinrich von Nagel.

(geometry) A triangle center defined as the point of concurrency of all three of the triangle's splitters.


equal detour point

equal detour point (plural equal detour points)

(geometry) A triangle center characterized by the property that, if one travels from any vertex of the triangle to another by taking a detour through some inner point, then the additional distance traveled is constant.


Mittenpunkt

German Mittenpunkt (literally “middle point”)

bor

(geometry) A triangle center defined as the symmedian point of the triangle formed by the centers of the three excircles.


Spieker circle

Named after 19th-century German geometer Theodor Spieker.

(geometry) The incircle of the medial triangle of a given triangle.


de Longchamps point

en-noun

de Longchamps point (plural de Longchamps points)

Named after French mathematician Gaston Albert Gohierre de Longchamps.

(geometry) A triangle center defined as the reflection of the orthocenter of the triangle about the circumcenter.


Napoleon point

The discovery of these points is sometimes attributed to Napoleon Bonaparte.

(geometry) Either of a pair of special points associated with a plane triangle. (See Wikipedia for detailed definitions.)

(geometry) Any of the isodynamic points of a triangle.


Gergonne triangle

Gergonne triangle (plural Gergonne triangles)

(geometry) A triangle defined by the three touchpoints of the incircle on the three sides of a given triangle.


Gergonne point

(geometry) A certain triangle center. Given a triangle ∆ABC and its Gergonne triangle (with the touchpoint opposite A denoted by T_A, etc.), the Gergonne point is the intersection of the three lines AT_A, BT_B, and CT_C.


scutoidal

en-adj

scutoidal (comparative more scutoidal, superlative most scutoidal)

more scutoidal (comparative)
most scutoidal (superlative)

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(geometry) Of, pertaining to, or having the shape of a scutoid.


Fourier-Mukai transform

en-noun

Fourier-Mukai transform (plural Fourier-Mukai transforms)

(algebraic geometry) A functor Φ_K between derived category of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y).


rectangular cuboid

rectangular cuboid (plural rectangular cuboids)

(geometry) A special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles.


Menelaus's theorem

(geometry) A proposition about triangles in plane geometry.