arcdegree

arcdegrees (plural)
arc degree (alternative)

/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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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.


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.


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 plane 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 vertices


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


star polyhedron

(geometry) A polyhedron that is partially nonconvex.


great cubicuboctahedron

en-noun

great cubicuboctahedron (plural great cubicuboctahedra)

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


small cubicuboctahedron

small cubicuboctahedron (plural small cubicuboctahedra)

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


isohedron

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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

(geometry) Synonym of intercept theorem.


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.


quindecile

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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.


superspatial

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Situated or taking place above or outside the limits of space.


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.


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


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


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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pursuit curve

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


inellipse

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(geometry) An ellipse inscribed in a triangle.


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 dodecahedron

en-noun

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}.


great icosahedron

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}.


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.


antinormal

(geometry) Perpendicular to a tangent of a curve or tangent plane of a surface but pointing opposite to convention.


antinormal

(geometry) A normal that points opposite to convention. For example, in solid geometry it is conventional for normals to point outwards, so an antinormal in solid geometry refers to a normal pointing inwards.


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.


hypersolid

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


hyperprism

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(geometry) A prism 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'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.


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.


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.


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.


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.


Tschirnhausen cubic

Tschirnhausen cubic (plural Tschirnhausen cubics)

Tschirnhausen cubics (plural)
Tschirnhaus' cubic (alternative)

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

hex a hec ta en ne a con ta kai hep ta gon

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.

triangle sphérique

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.


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).


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).


duocylinder

duocylinders (plural)
double cylinder (alternative)

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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.


goat grazing problem

goat grazing problem (plural goat grazing problems)

The problem: given a field of radius 1, and a goat tied to a point on the edge of said field; how long a rope should the goat be tied to, so that the goat has access to exactly half of the field? Equivalently: given a unit circle C and a point P on the rim of C, what is the value of r such that exactly half of the area within C, is within distance r of point P?

The same problem generalized to any number of dimensions: given a unit n-sphere (n-dimensional hypersphere) S, and a point P on the edge of S, what is r such that exactly one-half of the space (content) within S, is within distance r of P?


goat problem

goat grazing problem
goat grazing problem#English


direction vector

direction vector (plural direction vectors)

(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

semiminor axes (plural)
semi-minor axis (alternative)

(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.


Wallace line

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


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.


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.


elliptize

en-verb

elliptize (third-person singular simple present elliptizes, present participle elliptizing, simple past and past participle elliptized)

elliptizes (present singular third-person)
elliptizing (participle present)
elliptized (participle past)
elliptized (past)

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We resorted, instead, to depositing on a test flat near the mirror being elliptized.

1949, Albert Vinicio Baez, Principles of X-ray Optics and the Development of a Single Stage X-ray Microscope, Stanford University, page 132:

When a rotator cuff tear was present, the degenerative zone was elliptzed and typically a side-to-side repair was performed.

1991, Lonnie Paulos, James E. Tibone, editors, Operative Techniques in Shoulder Surgery, Aspen Publishers, page 86:

Note that least-square discretizations are also equivalent to making the convection terms elliptic; another way to ‘elliptize’ convection was derived from the Lax—Wendroff scheme by Lerat et al. (1982).

1996, Roger Peyret, editor, Handbook of Computational Fluid Mechanics, Academic Press, page 9:

The basic configuration was optimized for a 10 arcmin FoV at a Cassegrain focus issued from a four-segment spherical primary obtained by replication, a spherically polished secondary elliptized by partial vacuum (g=1) and a doublet-lens corrector.

1998, Larry M. Stepp, editor, Advanced Technology Optical/IR Telescopes VI, volume 3352, SPIE, page 767:

Thus a unified approach to both problems suggests itself. Following an idea of R. Picard (1985) we do it in such a way that one operator structures the null space of the other. Thus the operators mutually ‘elliptize’, simplifying the presentation.

2013 July, Rolf Leis, Initial Boundary Value Problems in Mathematical Physics, Dover Publications, page 170:

Although in almost all cases the orbits of the inner planets should remain within a certain range of distances from each other; in a "probability" estimated (in the models) at a few%, it is possible that the orbits elliptize enough to intersect and cause a catastrophic destabilization of the orbits of the inner planets.

2024 May, Georges Ballin, The Bacterium that will kill the Earth "the Man", [self-published], page 21:

(ergative, uncommon) (To cause) to become elliptic or elliptical.


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.


tangential triangle

en-noun

tangential triangle (plural tangential triangles)

A triangle formed of line segments tangential to the circumcircle of another triangle ABC at its vertices.


A polygon with 65537 sides and 65537 angles.


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.


obliquangled

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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 oblique-angled.


obtusangled

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(geometry) Obsolete form of obtuse-angled.


acutangled

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(geometry) Obsolete form of acute-angled.


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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 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.


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).


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.


Bevan circle

Named after British engineer Benjamin Bevan.

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


Bevan point

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


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 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.


Fermat-Torricelli point

Fermat-Torricelli point (plural Fermat-Torricelli points)


Torricelli point

Torricelli point (plural Torricelli points)


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.


Spieker center

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


Spieker circle

Named after 19th-century German geometer Theodor Spieker.

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


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.


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.


de Longchamps point

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.


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.


Gergonne triangle

en-noun

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.


rectangular cuboid

en-noun

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.


Carnot's theorem

(geometry) A theorem stating that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is DF + DG + DH = R + r, where r is the inradius and R the circumradius of the triangle.


conical surface

conical surface (plural conical surfaces)

(geometry) Synonym of cone (“a surface of revolution formed by rotating a segment of a line around another line that intersects the first line or a solid of revolution formed by rotating a triangle around one of its altitudes”).


triakis tetrahedron

triakis tetrahedron (plural triakis tetrahedra)

triakis tetrahedra (plural)
triakistetrahedron (alternative)

(geometry) A Catalan solid with 12 faces.


obpyramidal

Pyramidal, but having the apex downward; inversely pyramidal.


Loyd's paradox

(games, geometry) A paradox highlighting the slight changes in the arrangement of tangram pieces that lead to different visual perceptions and leading one to to believe that a piece is missing or extra.


morbus cyclometricus

(humorous, geometry) A fascination with finding a way to square the circle (an impossible task in geometry).


inverse Pythagorean theorem

the inverse Pythagorean theorem (canonical)

right-angled triangle
right-angled triangle

(geometry) A mathematical theorem which states that the square of the inverse length of the perpendicular bisector subtended by the right angle of a right-angled triangle is equal to the sum of the squares of the inverse lengths of the sides opposite the hypotenuse.


skew surface

(geometry) A ruled surface in which two successive straight lines do not intersect.