Riemannian geometry

Riemannian geometry (usually uncountable, plural Riemannian geometries)

From Riemannian + geometry, after German mathematician Bernhard Riemann.

Riemannian geometry was introduced by Riemann in 1854 as the n-dimensional generalization of the theory of curved surfaces of the 3D Euclidean space.

2001, Miklós Farkas, Dynamical Models in Biology, page 171:

Such geometries are called Riemannian geometries; they are characterized by invariant distances (for example, the space-time interval) that depend at most on the squares of the coordinate distances (∆x or ∆t).

2005, John F. Hawley, Katherine A. Holcomb, Foundations of Modern Cosmology, 2nd edition, page 235:

Riemannian geometry has found many applications in science, the most spectacular of these being the theory of relativity.[…]Every Riemannian geometry has geodesics, which are defined as the shortest curves joining two points.

2010, Saul Stahl, Geometry from Euclid to Knots, page 22:

At the beginning of the seventies, A. Gray generalized the classical holonomy concept by introducing a classification principle for non-integrable special Riemannian geometries [and] discovered in this context nearly Kähler manifolds in dimension six and nearly parallel G₂-manifolds in dimension seven.

2010, Ilka Agricola, “Chapter 9: Non-integrable geometries, torsion, and holonomy”, in Vicente Cortés, editor, Handbook of Pseudo-Riemannian Geometry and Supersymmetry, page 278:

The concepts of Riemannian geometry are familiar: length, angle, distance, and curvature, among others. Historically tied to the origins of differential geometry, and with such familiar concepts, Riemannian geometry is often presented in textbooks as being synonymous with differential geometry itself, instead of as one differential-geometric structure among many.

2013, Andrew McInerney, First Steps in Differential Geometry: Riemannian, Contact, Symplectic, page 195:

(mathematics, geometry) The branch of differential geometry that concerns Riemannian manifolds; an example of a geometry that involves Riemannian manifolds.


fourgon

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(geometry) A polygon with four sides; a quadrilateral.


square the circle

en-verb

square the circle (third-person singular simple present squares the circle, present participle squaring the circle, simple past and past participle squared the circle)

squares the circle (present singular third-person)
squaring the circle (participle present)
squared the circle (participle past)
squared the circle (past)

The problem of squaring the circle is the problem of determining the ratio of the circumference of a circle to the diameter, which is called π.

1946, Bertrand Russell, History of Western Philosophy, I.24:

(geometry) To construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge, a task which is mathematically impossible.

رَبَّعَ الدائِرة

to construct a square with the same area as a circle

square peg in a round hole

semantic overlap regarding a difficult problem


intersphere

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(geometry) A sphere that is tangent to every edge of a given polyhedron.


subcontrary

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Two planes drawn through the vertex of the cone parallel to the plane of the circular base , and that of a subcontrary section, are called by Chasles the cyclic planes of the cone

1852, John Mulcahy, Principles of Modern Geometry:

(geometry) Having, or being in, a contrary order; said of a section of an oblique cone having a circular base made by a plane not parallel to the base, but so inclined to the axis that the section is a circle; applied also to two similar triangles when so placed as to have a common angle at the vertex, the opposite sides not being parallel.


subtense

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(geometry) A line which subtends, especially the chord of an arc.

(geometry) The angle subtended by a line at a point.


trihedral

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(geometry) Having three plane faces that meet at a common point


contingence

en-noun

contingence (countable and uncountable, plural contingences)

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Borrowed from Latin contingentia, from contingēns, present passive participle of contingō (“I make contact with, I am touching”).


isohedral

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(geometry) Describing a polyhedron or a tiling all of whose faces are the same.


isodynamic

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From Ancient Greek ἰσοδῠ́νᾰμος (isodŭ́nămos, “equal in power”) + -ic, after dynamic. By surface analysis, iso- + dynamic.

affix

grc:ἰσοδῠ́νᾰμος

equilateral triangle
equilateral triangle

inversely proportional
inversely proportional

(geometry) In Euclidean geometry, being or relating to certain points associated with a triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle.


conchoid

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From Latin concha (“mussel”) (from Ancient Greek κόγχη (kónkhē)) + -oid or directly from Ancient Greek κογχοειδής (konkhoeidḗs), referring to the curved outline of a mussel shell or anything in this form.

The conchoid of a circle with respect to a point on the circle is a cardioid if the fixed distance is equal to the diameter of the circle.

The Conchoid of Nicomedes is the conchoid of a straight line with respect to a point not on the line.

1815, Charles Hutton, Pappus, entry in A Philosophical and Mathematical Dictionary, Volume 2, page 147,
He next treats of the properties of the Conchoid, which Nicomedes invented for doubling the cube; applying it to the solution of certain problems concerning Inclinations, with the finding of two mean proportionals, and cubes in any proportion whatever.

The classical conchoid construction is a non-orthogonal polar-curvilinear construction in which equal distances along a line are marked off from its point of intersection with a curve for various positions of the line as it rotates about a point.

1982, J. Lee Kavanau, Curves and Symmetry, volume 1, page 3:

These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell.

2007, James Stewart, Single Variable Calculus, volume 2, page 662:

One of the best choices is the conchoid, according to Newton the simplest curve after the circle.

2009, Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method, page 68:

(mathematics, geometry) Any of a family of curves defined as the locus of points p, such that each p is on a line that passes through a given fixed point P and intersects a given curve, C, and the distance from p to the point of intersection with C is a specified constant (note that for nontrivial cases two such points p satisfy the criteria, and the resultant curve has two parts).


hypercone

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(geometry) A four-dimensional surface related to the three-dimensional cone


prismoid

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(geometry) A prismatoid that has planar sides, and the same number of vertices in both of its parallel planes.


hyperbolism

en-noun

hyperbolism (countable and uncountable, plural hyperbolisms)

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(countable, geometry) A blow-up of a curve at a point.


semisquare

semisquares (plural)
semi-square (alternative)

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(geometry) A right isocoles triangle, or such figure of which two form a square.


arbelos

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From Ancient Greek ἄρβηλος (árbēlos) or ἄρβυλος (árbulos), literally "shoemaker's knife".

(geometry) A plane region bounded by a semicircle of diameter 1, connected to semicircles of diameters r and (1 − r), all oriented the same way and sharing a common baseline.


excircle

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Also since the circle of inversion cuts both excircles orthogonally, each excircle inverts into itself.

1979, Dan Pedoe, Circles: A Mathematical View, published 1995, page 10:

Extend the sides of triangle QRS and construct the three excircles: One excircle is tangent to side QR and rays SQ and SR; one excircle is tangent to side SR and rays QS and QR; and one excircle is tangent to side SQ and rays RS and RQ.

1999, Art Johnson, Famous Problems and Their Mathematicians, Teacher Ideas Press, page 174:

Lemma 4.9 (The Diameter of the Incircle). Let ABC be a triangle whose incircle is tangent to #92;overline#123;BC#125; at D. If #92;overline#123;DE#125; is a diameter of the incircle and ray AE meets #92;overline#123;BC#125; at X, then BD#61;CX and X is the tangency point of the A-excircle to #92;overline#123;BC#125;.
Incircles and excircles often have dual properties.

2016, Evan Chen, Euclidean Geometry in Mathematical Olympiads, page 61:

(geometry) An escribed circle; a circle outside a polygon (especially a triangle, but also sometimes a quadrilateral) that is tangent to each of the lines on which the sides of the polygon lie.


salinon

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From Ancient Greek *σάλινον (*sálinon, unattested), presumably from and meaning the same as Latin salīnum (“salt-cellar”), derived from sal (“salt”) (compare salillum).

bor

(geometry) A particular figure bounded by four semicircles.


hemiequilateral

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(geometry) Describing the shape of a triangle formed by cutting an equilateral triangle in half through one of the apices


describent

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Latin dēscrībēns, present participle of dēscrībō (“I describe”).


evolvent

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Latin ēvolvēns, ēvolventis (“unrolling”), present participle of ēvolvō (“I unroll”).


equidecomposable

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(geometry, of two forms) Capable of being decomposed into a number of pieces that can be reassembled into either of the two.


equidecomposability

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(geometry) The quality of being equidecomposable.


isocapacitary

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(geometry) Having the same capacity; applied to the problem of finding an extremum for an integral over a space of a specified capacity.


capacitary

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horocyclic

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In two-dimensional Euclidean geometry using a polar coordinate system, horocyclic flow is either the circular convergence toward or away from a point or the rotational flow around a point.

Hence the regular polygon ABCD ... can either be a convex n-gon, a star n-gon, a horocylic apeirogon or a hypercyclic apeirogon.

1984, Coxeter-Festschrift [Mitteilungen aus dem Mathem[atisches] Seminar Giessen], Giessen: Gießen Mathematischen Institut, Justus Liebig-Universität Gießen, page 247:

(geometry) Relating to a set of points that are all equidistant (keeping other coordinates constant) from some constant point.


en-adj

3-dimensional (comparative more 3-dimensional, superlative most 3-dimensional)

more 3-dimensional (comparative)
most 3-dimensional (superlative)


4-dimensional (comparative more 4-dimensional, superlative most 4-dimensional)

more 4-dimensional (comparative)
most 4-dimensional (superlative)


2-dimensional (comparative more 2-dimensional, superlative most 2-dimensional)

more 2-dimensional (comparative)
most 2-dimensional (superlative)


rhombicuboctahedron

en-noun

rhombicuboctahedron (plural rhombicuboctahedra or rhombicuboctahedrons)

rhombicuboctahedra (plural)
rhombicuboctahedrons (plural)

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From Latin rhombicuboctahaedron,
referring to the fact that the twelve of the square faces lie in the same planes as the twelve faces of the rhombic dodecahedron, the dual of the cuboctahedron. (See rhombicuboctahedron on Wikipedia.Wikipedia)

Therefore eight triangles and eighteen (that is, twelve and six) squares join up to make an icosihexahedron, which I call a truncated cuboctahedral rhombus or a rhombicuboctahedron.

1997, Johannes Kepler, E. J. Aiton, Alistair Matheson Duncan, Judith Veronica Field, transl., The harmony of the world, American Philosophical Society, page 119:

The gaps in a face-centered cubic packing of square sharing rhombicuboctahedra can be filled by cubes and tetrahedra (Fig. 2.4(h)).

2009, Walter Steurer, Sofia Deloudi, Crystallography of Quasicrystals: Concepts, Methods and Structures, page 55:

So, for instance, covering up the right-hand side of our example (as in Figure 20.2(a)), we see that the rhombicuboctahedron does indeed have three types of face: two four-sided ones and one three-sided one.

2016, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, page 273:

small rhombicuboctahedron
small rhombicuboctahedron

(geometry) An Archimedean solid with eight triangular and eighteen square faces; the small rhombicuboctahedron.

rombikuboktaedri

polyhedron with 8 triangular and 18 square faces

斜方立方八面体

しゃほうりっぽうはちめんたい

菱形立方八面体

りょうけいりっぽうはちめんたい


SDG

SDG (countable and uncountable, plural SDGs)

synthetic differential geometry
w:synthetic differential geometry

(geometry) Initialism of synthetic differential geometry.


geom.

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coöriginal

/ˌkəʊəˈɹɪdʒənəl/

/ˌkoʊəˈɹɪd͡ʒənəl/

co-original (alternative)
cooriginal (alternative)

In operations on lines in space, it is convenient to substitute for some of the given lines other lines, equal, and therefore parallel, to the given ones, but passing through a point in space common to themselves and the other given lines; so that all the lines may be coöriginal.

1859 October (pub. 1860 October), John Daniel Runkle (editor), The Mathematical Monthly, volume II, № I, pages 29–31: W.P.G. Bartlett, “The Elements of Quaternions”, part I: “Lines”, § 4

(geometry, of a line) Originating from the same point in space.


hendecahedron

hendecahedron (plural hendecahedrons or hendecahedra)

hendecahedrons (plural)
hendecahedra (plural)

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(geometry) A polyhedron with eleven faces.


coterminal

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When you try to draw two coterminal angles on a circle, you will find that you only get one line.

(geometry, of two angles) Differing only by a whole number of complete circles.

coterminal

angle differing only by a whole number of complete circles


oblongum

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(geometry, archaic) A prolate spheroid; a figure described by the revolution of an ellipse about its greater axis.


hemicube

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(geometry) An abstract regular polyhedron containing half the faces of a cube.


cross-ratio

cross-ratios (plural)
crossratio (alternative)

(geometry) A special number associated with an ordered quadruple of collinear points, particularly points on a projective line, and preserved by the fractional linear transformations.

birapport

a special number associated with an ordered quadruple of collinear points, particularly points on a projective line, and preserved by the fractional linear transformations


explementary angles

(geometry) A pair of angles that sum to 360 degrees.


conjugate angles

(geometry) A set of angles that sum to 360 degrees.


Cassini oval

Named after the astronomer Giovanni Domenico Cassini, who studied them in 1680.

(geometry) A plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant (related to an ellipse, for which the sum of the distances is constant).


psc

(geometry) Abbreviation of parallax second.


quadrantal

From Latin quadrantālis (“quarter, containing the fourth part of a measure”).

der

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quadrantal space; quadrantal triangle; quadrantal versor

(geometry) Of or relating to a quadrant, or included in the fourth part of a circle.


achirality

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(geometry, physical chemistry) The characteristic of being achiral; the absence of chirality.


trihexagonal

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(geometry) Pertaining to a semiregular tiling of the Euclidean plane with two triangles and two hexagons alternating on each vertex.


hexoctahedron

en-noun

hexoctahedron (plural hexoctahedrons or hexoctahedra)

hexoctahedrons (plural)
hexoctahedra (plural)
hexaoctahedron (alternative)

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(geometry) A solid with forty-eight equal triangular faces.


multilobe

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oxygon

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enneagonal

enneagonal angle

(geometry) In the form of an enneagon; having nine angles.


curvilinead

(geometry, obsolete) An instrument for drawing curved lines.


orthotomy

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(geometry) The property of cutting at right angles.


trirectangular

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retrosnub

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great retrosnub icosidodecahedron

small retrosnub icosicosidodecahedron

(geometry) Derived from a simpler, regular polyhedron by the addition of extra triangular faces in the back.


excentral

en-adj

excentral (comparative more excentral, superlative most excentral)

more excentral (comparative)
most excentral (superlative)

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scalenohedron

en-noun

scalenohedron (plural scalenohedrons or scalenohedra)

scalenohedrons (plural)
scalenohedra (plural)

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Though the hexagonal scalenohedron is derived from the double twelve-faced pyramid, by the development of half its faces, and might be constructed from that figure, it is more readily obtained from the positive or negative rhomboid which may be supposed to be inscribed in the scalenohedron.

1859, James Tennant, Walter Mitchell, Mineralogy and Crystallography: Being a Classification of Crystals According to Their Form, page 408:

By bringing together the rhombohedron and the scalenohedron, as has been done in Fig. 119, it will be noticed that the lateral edges of the two forms have a similar position towards the axes, so that for every scalenohedron there must be a rhombohedron whose lateral edges coincide with the lateral edges of the other form. This rhombohedron is called the inscribed rhombohedron of the scalenohedron.

1866, Josiah Parsons Cooke, Elements of Chemical Physics, page 154:

1907, Benjamin K. Emerson, Joseph H. Perry, The Green Schists and Associated Granites and Porphyries of Rhode Island, US Geological Survey, Bulletin Number 311, page 24,
Other steep scalenohedrons of the principal zone are also often present, sometimes in addition to y, sometimes taking its place.

Calcite crystals show a great variety of habits — acute and obtuse rhombohedra and scalenohedra, prisms, and thin tabular forms.

1940, John W. Evans, George MacDonald Davies, Elementary Crystallography, page 111:

(geometry, mineralogy, crystallography) A polyhedron having twelve sides, each in the form of a scalene triangle, that is topologically equivalent to a hexagonal bipyramid and whose middle section can be said to inscribe a rhombohedron.


subtangent

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(geometry) The part of the axis contained between the ordinate and tangent drawn to the same point in a curve.


cissoid

From Ancient Greek κισσός (kissós, “ivy”) + εἶδος (eîdos, “form, image”).

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The cissoid of a circle and a line is a conchoid

(geometry) Any of a family of curves defined as the locus of a point, P, on a line from a given fixed point and intersecting two given curves, C₁ and C₂, where the distance along the line from C₁ to P remains constant and equal to the distance from P to C₂.


semitangent

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crunodal

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crunodal curve

crunodal cubic

(geometry) Possessing, or characterized by, a crunode.


amblygon

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From Ancient Greek ἀμβλύς (amblús) + -gon.

(geometry, obsolete, rare) An obtuse-angled shape, especially an obtuse triangle.


cyclometry

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(archaic, geometry) The measurement of circles, including calculating circumference, arcs, angles etc.


triquadrantal

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A triquadrantal triangle is one whose three sides are quadrants, and whose three angles are consequently right angles.


spheroconic

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(geometry) A non-plane curve formed by the intersection of the surface of an oblique cone with the surface of a sphere centred at the cone's vertex.


cylindroid

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The intrados of a simple vault is generally formed of the portion of the surface of a cylinder, cylindroid, or sphere, never greater than that of half the solid; and the springing lines of the coussinet, which terminate the walls from which the vault rises, are generally straight lines, parallel to the axis of the cylinder or cylindroid.

1830, Sir David Brewster, The Edinburgh Encyclopaedia, page 523:

(geometry) A cylinder with elliptical cross-section.

(geometry) A certain three-dimensional surface described by a moving straight line, used to illustrate the motions of a rigid body and the forces that act upon it.


symmetral

Partly from Latin symmetros and partly from Ancient Greek σύμμετρος (súmmetros), + -al.


orthotomic

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orthotomic

(geometry) The line that joins the poles of the common chord on two orthotomic circles.

(geometry) The envelope of the family of circles that have centers on a specified curve and which pass through a specified point.


Steiner point

(geometry) The point on the circumcircle of a triangle that lies at the intersection of three lines through the vertices of the triangle, where those lines are parallel to the corresponding sides of the first Brocard triangle.

(geometry) The geometric centroid of the system obtained by placing a mass equal to the magnitude of the exterior angle at each vertex of the system.

(geometry) Any of the 20 points located at the 3 by 3 intersections of the 60 Pascal lines which are produced by extending the triplets on opposite sides of a conic section in Pascal's theorem for all permutations of verices.


bimedial

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(geometry, of a line) Being the sum of two lines commensurable only in power (such as the side and diagonal of a square).


interaxis

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The principles of analytical geometry afford the means of determining how much the interaxes of the regular octahedron must be increased or diminished to equal the interaxes in these different forms.

1854, James Dwight Dana, A System of Mineralogy, Comprising the Most Recent Discoveries, page 106:

Owing to this I was only able, by breaking a number of the quartz pieces, to obtain one small crystal representing a distorted rhombic dodecahedron ∞ 0(110), i.e., one extended in the line of an octahedral or trigonal interaxis, giving it the appearance of an hexagonal prism terminated by a rhombohedron (see fig. 2) in striking resemblance to dioptase.

1885, George H. F. Ulrich, “Notes on some Mineral Occurrences At Dusky Sound, West coast of Middle Island, New Zealand”, in The New Zealand Journal of Science, volume 1, page 309:

When the position is as given in the stereographic projection in Fig. 265, the angle over the axial-plane polar edges is the more acute, but if the pole were placed nearer to the crystallographic than to the interaxis, as in the case of the dihexagonal pyramid {2131} of beryl shown in Fig . 266, the case would be reversed.

1911, Alfred Edwin Howard Tutton, Crystallography and Practical Crystal Measurement, page 306:

In the traditional valence bond approach above, the decrease of the bron position parameter improves the bond strength of the (sp³)_z hybrid, but it increases the misfit between direction of the boron-metal interaxis and direction (z-axis) of (sp³)_z hybrid.

1968, Forrest L. Carter, “Valence Bonding in Some Refractory Transition Metal Compounds with High Coordination”, in NBS Special Publication, page 548:

(geometry) A line from the center of a regular solid to the center of one of its edges.


amblygonal

en-adj

amblygonal (comparative more amblygonal, superlative most amblygonal)

more amblygonal (comparative)
most amblygonal (superlative)

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osculatrix

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From New Latin osculātrīx (literally “female kisser”), from Latin osculor (“to kiss”) + -trīx.

bor

(geometry) A curve whose contact with a given curve at a given point is of a higher order (or involves the equality of a greater number of successive differential coefficients of the ordinates of the curves taken at that point) than that of any other curve of the same kind.


radius vector

en-noun

radius vector (plural radii vectores or radius vectors)

radii vectores (plural)
radius vectors (plural)

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


tractory

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Latin tractorius (“of drawing”), from trahere, tractum (“to draw”).


bicircular

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a bicircular curve

(geometry) Having nodes at the two circular points at infinity.


anticomplementary

en-adj

anticomplementary (comparative more anticomplementary, superlative most anticomplementary)

more anticomplementary (comparative)
most anticomplementary (superlative)

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semiparabola

en-noun

semiparabola (plural semiparabolas or semiparabolae)

semiparabolas (plural)
semiparabolae (plural)

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(geometry) One branch of a parabola, terminating at the principal vertex of the curve.


osculant

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From Latin osculans, osculantis, present participle of osculari (“to kiss”). See osculate.

(geometry) Tangent, touching at a single point.


osculant

(geometry) The point at which two tangent curves touch.


trisoctahedron

trisoctahedron (plural trisoctahedrons or trisoctahedra)

trisoctahedrons (plural)
trisoctahedra (plural)

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subcardioid

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(geometry) Partially or imperfectly cardioid.


septangle

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crunode

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Probably from Latin crux (“a cross”) + English node.

(geometry, archaic) node (point where one branch of a curve crosses another branch)


Didonia

In allusion to the classical story of Queen Dido, who was granted the land she could encompass with a piece of bull hide.

And hence, by Meusnier's theorem, the difference of the squares of the curvatures of curve and surface is constant; the curvature of the surface meaning here the reciprocal of the radius of the sphere which osculates in the reduction of the element of the Didonia.

1867, Peter Guthrie Tait, An Elementary Treatise on Quaternions:

(geometry) The curve which on a given surface and with a given perimeter contains the greatest area.


semiangle

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(geometry) an angle of half a given size


nodated

en-adj

nodated (comparative more nodated, superlative most nodated)

more nodated (comparative)
most nodated (superlative)

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From Latin nodatus, past participle of nodare (“to make knotty”), from nodus (“knot”). See node.

Form A to b or B to b, along the circular arc (see fig. 4), the paths of a series of points will give a gradation of symmetrical double nodated figures between the arc of a circle and the cuspidated tirangular figure. This is the first or finite series of symmetrical double nodated lines.

1830, The Mechanics' Magazine, Museum, Register, Journal, and Gazette, page 232:


tractricoid

(geometry) A surface formed by revolving a tractrix about its asymptote.


striction

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(geometry) The definition of a geometric shape as all points with some relationship to a central point, curve, etc., for example, a circle is the striction of all points at a distance of the radius from a single point.


line of striction

en-noun

line of striction (plural lines of striction)

(geometry) The line on a skew surface that cuts each generator in that point of it that is nearest to the succeeding generator.


orthotope

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(geometry) The generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.


hyperrectangle

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semilune

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pentacron

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From Ancient Greek πέντε (pénte, “five”) + ἄκρον (ákron, “summit”).

(archaic, geometry) A solid with five summits (vertices), such as a square pyramid


sexangle

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From Latin sexangulus (“sexangular”), from sex (“six”) + angulus (“angle”). Compare French sexangle and English hexangular; sex- + angle.

the super parts being taken from the Quadrangle will form the Sexangle

1828, James Wilson (translator), Claudius Ptolomaeus, The Tetrabiblos:


cyclohedron

(geometry) A certain (n−1)-dimensional polytope that is useful in studying knot invariants.


covertex

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(geometry) A point of intersection between an ellipse and its minor axis.


bialtitude

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(geometry) A perpendicular to two opposite edges of a tetrahedron.


orthocentric

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(geometry, of a tetrahedron) Having all three pairs of opposite edges perpendicular.

(geometry) Of or relating to a set of four points in the plane, one of which is the orthocenter of the triangle formed by the other three.


dextrorse

en-adj

dextrorse (comparative more dextrorse, superlative most dextrorse)

more dextrorse (comparative)
most dextrorse (superlative)

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From Latin dextrorsum, contracted from dextrovorsum, dextroversum (“toward the right side”), from dexter (“right”) + versus, vorsus, past participle of vertere, vortere (“to turn”).


paracentric

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(geometry) A planar curve having the property that, when its plane is placed vertically, a body descending along it, by the force of gravity, will approach to, or recede from, a fixed point or centre, by equal distances in equal times.


apotome

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Apotome
Euclid's Elements
Pythagoreanism#Music

Borrowing from Ancient Greek ἀποτομή (apotomḗ, “cutting off”). The musical sense originates from the Pythagorean tradition. The mathematical sense is attested in Euclid's Elements (Book X, proposition 73, et seq.).

bor

Yet it is not until Book X that the properties of such a line (with greater length is an apotome and lesser length a first apotome) are explained and not until Book XIII that this type of line is applied, which will be discussed in more detail later.[…]The likewise is true of apotomes (X. 97).

2012, Jade Roskam, “Book X of The Elements: Ordering Irrationals”, in Bharath Sriraman, editor, Crossroads in the History of Mathematics and Mathematics Education, Information Age Publishing, page 210:

2014, Jacques Sesiano (translator), Liber Mahameleth, Part Two: Translation, Glossary, [12th c, Anonymous (possibly John of Seville), Liber Mahameleth], Springer, page 767,
If some number and the root of the root of a number are multiplied by the corresponding apotome, the result will be an apotome.

(mathematics, geometry) The difference between two quantities or lengths commensurable only in power, as between 1 and the square root of 2, or between the diagonal and side of a square.

ἀποτομή

difference between two quantities commensurable only in power


homaloidal

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Ancient Greek ὁμαλός (homalós, “even, level”) + + -oid.

uder

(geometry) flat; even; applied to surfaces and spaces in which the definitions, axioms, and postulates of Euclid respecting parallel straight lines are assumed to hold true.


hypervolume

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(geometry) A volume in more than three dimensions.


enneahedron

en-noun

enneahedron (plural enneahedrons or enneahedra)

enneahedrons (plural)
enneahedra (plural)

The enneahedron is a simple all-space filler discovered by myself, based on the cubic antiprism, again omitted by Wood.

1968, Design, numbers 229-234, unnumbered page:

1996, Michael O'Keeffe, Bruce G. Hyde, Crystal Structures, Number 1: Patterns & Symmetry, page 350,
This is based on a packing of 36-vertex icosahedra [5¹².6⁸] and enneahedra (4³.5⁶].

In Figure 9.10.3 we see a projection of the skeleton of an enneahedron (a solid with nine faces). All the faces of this enneahedron are quadrilaterals.

2015, Claudi Alsina, Roger B. Nelsen, A Mathematical Space Odyssey: Solid Geometry in the 21st Century, page 191:

(geometry) A polyhedron with nine faces.


parallelotope

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Blend of parallelogram + polytope. Popularized by Canadian geometer H.S.M. Coxeter.

1983 [North Holland/Elsevier], Robert S. Borden, A Course in Advanced Calculus, 1998, Dover, page 83,
Suppose now that P is a k-dimensional parallelotope, 1<k<n, in Eⁿ.

1997, Daniel A. Klain, Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge University Press, Accademia Nazionale dei Lincei, page 30,
We develop next a theory of invariant valuations for the lattice of finite unions of orthogonal parallelotopes, having edgers parallel to a fixed frame. […] This follows from the basic fact that the intersection of two parallelotopes is a parallelotope.

It has 2ᵏ gyrovertices, k#43;1 of which are the gyropoints of S, and k 2#123;k-1#125; gyroedges, in full analogy with parallelotopes.

2015, Abraham Albert Ungar, Analytic Hyperbolic Geometry in N Dimensions, Taylor & Francis (CRC Press), page 187:

(geometry) A generalization of parallelogram and parallelepiped into a figure of arbitrary dimensionality.

Harold Scott MacDonald Coxeter
parallelotope


projective plane

projective plane (plural projective planes)

Generally, a projective plane of "order" N has N²+N+1 points, and the same number of lines.

2004, David S. Dummit with Richard M. Foote, chapter 6, in Abstract Algebra, page 210

(geometry) A plane in which any pair of distinct points determine a line and any pair of distinct lines intersect at a point.


pentamery

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(geometry) the condition of being pentameric


trimery

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(geometry) The condition of being trimeric.


supercatenoid

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(algebra, geometry) a geometric surface constituting a transition between a cube (or similar surface) and a catenoid.


superegg

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(geometry) A surface formed by rotating a superellipse about one of its axes.


supershape

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(geometry) Any closed curve that is a generalization of a superellipse; it is defined by a superformula


superquadric

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(geometry) A three-dimensional relative of a superellipse (or similar curve).


porism

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From Ancient Greek πόρισμα (pórisma, “a deduction from a previous demonstration”).

der

(geometry, number theory) A proposition affirming the possibility of finding such conditions as will render a certain determinate problem indeterminate or capable of innumerable solutions.


mixtilinear

en-adj

mixtilinear (comparative more mixtilinear, superlative most mixtilinear)

more mixtilinear (comparative)
most mixtilinear (superlative)
mixtilineal (alternative)

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Latin mixtus (“mixed”) + English linear.

A mixtilinear angle is an angle contained by a straight line and a curve.

(geometry) Containing, or consisting of, lines of different kinds (straight, curved, etc.).


explementary

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Two angles whose sum is a circle (360°) are explementary angles, and either is the explement of the other.

2002, Nathaniel Bowditch, American Practical Navigator, page 317:

(geometry) Of or relating to the explement


hyperface

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(geometry) A hyperplane that forms a boundary of a polytope


rhombille

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heterocentric

heterocentric (comparative more heterocentric, superlative most heterocentric)

more heterocentric (comparative)
most heterocentric (superlative)

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The purpose of this paper is to develop a complete and general theory of aperture referral in linear optics for dioptric systems that may be heterocentric and astigmatic. Apertures may be elliptical and decentred. Referral transforms the geometry of an aperture.

2011, William F. Harris, “Aperture referral in heterocentric astigmatic systems”, in Ophthalmic and Physiological Optics, volume 31, number 6, →DOI, →ISSN, page 603:

(geometry) Having different or multiple centres.


selective service
w:Selective Service System

A selective service class denoting that the registrant is already a reserve member of the armed forces or a student taking military training, as in ROTC or an accepted aviation cadet applicant.


catenarian

catenarian (comparative more catenarian, superlative most catenarian)

more catenarian (comparative)
most catenarian (superlative)

The curve so formed is a catenarian curve, and, when inverted, delineates the curve of a type of arch better able to resist forces of destruction than the earlier semicircular arch. Investigators who followed Galilei and studied the catenarian arch mathematically were able to show that a simple catenarian outline was formed by the chan suspeneded as already described, the length of the chain depending upon the required span and rise of the arch.

1957, Bernard E. Jones, Freemason's Book of the Royal Arch, page 105:

a catenarian curve/arch

(geometry) Shaped like a catenary; following the curve made by a chain or rope when suspended from both ends.


heptacontagon

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(geometry) A polygon with seventy edges.


tetracontagon

Blend of tetra- (“4”) + Ancient Greek τεσσαράκοντα (tessarákonta, “40”) + -gon (“-sided”), from γωνία (gōnía, “corner”).

der

τεσσαράκοντα


pedal curve

(geometry) For a plane curve C and a given fixed point P, the locus of points X such that PX is perpendicular to a tangent to the curve passing through X.


In every vertex of the 5-cell, 8-cell and 120-cell there are 4 concurring edges; in every vertex of the 24-cell, 8 edges.

1896, American Journal of Mathematics, page 181:

1988, Unnamed translator, L. A. Sidorov, Confuguration, article in Michiel Hazewinkel (editor) Encyclopaedia of Mathematics: An updated and annotated translation of the Soviet ‘Mathematical Encyclopaedia’, Volume 2: C, page 309,
For example, a 5-cell is bounded by five three-dimensional tetrahedra, an 8-cell by eight cubes, etc., the 5-cell and the 24-cell being duals of each other (the points correspond to spaces and lines to planes).

If we call a regular polytope with m facets an m-cell, then in E⁴ we have the 5-cell with tetrahedral facets (the simplex), the 8-cell with cubic facets (the hypercube), the 16-cell with tetrahedral facets (the regular cross-polytope), the 24-cell with octahedral facets, the 120-cell with dodecahedral facets, and the 600-cell with tetrahedral facets.

2007, Steven R. Lay, Convex Sets and Their Applications, page 230:

Later in the same document, President Hall listed all of the university's four-dimensional models: four Brill models—the 5-cell, 8-cell, 16-cell, and 24-cell—along with seven spun glass models by T. P. Hall that "illustrated rotations."

2008, Tony Robbin, Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought, page 16:

(geometry) A four-dimensional polytope, analogous to a tetrahedron, whose five bounding facets are tetrahedra.

kvinĉelo

four-dimensional polytope analogous to a tetrahedron

4-dimensional polytope analogous to a tetrahedron


hexadecachoron

en-noun

hexadecachoron (plural hexadecachorons or hexadecachora)

hexadecachorons (plural)
hexadecachora (plural)

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From hexadeca- (“sixteen”) + -choron (“room”), from Ancient Greek ἕξ (héx, “six”), δέκα (déka, “ten”) and χῶρος (khôros, “room”).

confix

In a world with four spatial dimensions, for example, we can construct only six regular solids, viz pentatope, tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron and hexacosichoron.

2005, V H Satheesh Kumar, P K Suresh, Are We Living in a Higher Dimensional Universe?, page 3:

We have chosen the surfaces of a pentatope (5-cell) as well as a hexadecachoron (16-cell), which are simple standard triangulations of a 3-sphere, shown in figure (7), as our underlying lattices.

2009, P. Khavari, C. C. Dyer, Aspects of Causality in Parallelisable Implicit Evolution Scheme, page 7:

In four dimensions, surprisingly, there are three space-filling convex regular polychora: the tesseract (the hypercube in ℝ⁴), the hexadecachoron, and the icositetrachoron.

2011, Jin Akiyama, Ikuro Sato, The element number of the convex regular polytopes, page 271:

(mathematics) A four-dimensional object analogous to an octahedron, constructed out of sixteen tetrahedra.


The remaining 8, as before, are the vertices of the complementary 16-cell (see Figure 9). In Figure 7, these 8 points have been blackened, but not joined, because the 16-cell has longer edges; diameters of octahedral facets of the 24-cell.

1995, Harold Scott Macdonald Coxeter, “Paper Three: Two Aspects Of The Regular 24-Cell In Four Dimensions”, in F. Arthur Sherk, editor, Kaleidoscopes: Selected Writings of H.S.M. Coxeter, page 32:

This polytope has 16 tetrahedral cells, and for this reason it is called the 16-cell.[…]The 16-cell has 24 edges that are the diagonals of the 24 square faces of the hypercube. (Since we are selecting alternate vertices of the hypercube, we only get one diagonal for each square face.)

2002, Gabor Toth, Glimpses of Algebra and Geometry, 2nd edition, page 385:

We will now consider the dual 16-cell. In some ways the 16-cell is a bit easier because it has fewer vertices and edges. But on the other hand the hypercube is perhaps easier to understand intuitively because we can readily (perhaps) see the various cubes of which it is composed.

2012, John Barnes, Gems of Geometry, 2nd edition, page 76:

(geometry) A four-dimensional polytope, analogous to an octahedron, whose sixteen bounding facets are tetrahedra.

4-dimensional polytope analogous to an octahedron


octachoron

en-noun

octachoron (plural octachorons or octachora)

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

From octa- (“eight”) + -choron (“room”), from Ancient Greek ὀκτά- (oktá-, “eight”) and χῶρος (khôros, “room”).

Our 16-point 4D constellation is described by the set {(±1,±1,±1,±1)}, which is indeed the set of vertices of a tesseract (i.e., a regular octachoron) [4].

2012, Yequn Zhang, Murat Arabaci, Ivan B. Djordjevic, Rate-Adaptive Four-Dimensional Nonbinary LDPC-Coded Modulation for Long-Haul Optical Transport Networks, page 3:

The Schlegel diagram from this figure is a projection of a regular octachoron from the 4-D space to the 3-D space through a point beyond one of its facets.

2013, Milorad Cvijetic, Ivan B. Djordjevic, Advanced Optical Communication Systems and Networks, page 357:

...completing the notation for the vertices of an elementary lattice octachoron.

2013, Paul Jennings, Frank Nijhoff, On an elliptic extension of the Kadomtsev-Petviashvili equation, page 4:

(mathematics) A four-dimensional object analogous to a cube, constructed out of eight cubes.


According to this table the tetrahedron corresponds to the 5-cell, the cube to the 8-cell ("hypercube"), the octahedron to the 16-cell, the dodecahedron to the 120-cell and the icosahedron to the 600-cell. The 24-cell is an additional body in four-dimensional space which is dual to itself.

1988, Klaus Mainzer, Symmetries of Nature: A Handbook for Philosophy of Nature and Science, published 1996, page 159:

Similarly in dimension eight there is the quaternionic projective plane which we can think of as a 4-sphere with an 8-cell attached, and in dimension sixteen there is a Cayley projective plane which has similar properties.

1996, John Milnor, “Growing Up in the Old Fine Hall”, in Hugo Rossi, editor, Prospects in Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, page 6:

Of these nine, five are finite structures in Euclidean space and include the simplex, {3,3,3} (or 5-cell), the four-dimensional cube {4,3,3} (or 8-cell) and its dual {3,3,4} (or 16-cell), and the 120-cell {5,3,3} and its dual {3,3,5} (or 600-cell).

1997, J. François Gabriel, editor, Beyond the Cube: The Architecture of Space Frames and Polyhedra, page 412:

(geometry) A four-dimensional polytope, analogous to a cube, whose eight bounding facets are cubes; a tesseract.

4-dimensional polytope analogous to a cube


icositetrachoron

icositetrachoron (plural icositetrachorons or icositetrachora)

icositetrachorons (plural)
icositetrachora (plural)

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From icosi- (“twenty”) + tetra- (“four”) + -choron (“room”), from Ancient Greek εἴκοσι (eíkosi, “twenty”) τέτταρες (téttares, “four”) and χῶρος (khôros, “room”).

There are six convex regular 4-polytopes; five of these are the higher dimensional analogs of the Platonic solids, and one additional figure, the icositetrachoron, has no three-dimensional equivalent.

2011, Jin Akiyama, Ikuro Sato, The element number of the convex regular polytopes, page 270:

The icositetrachoron (hyperdiamond, octaplex or 24-cell) is the most unique polychoron, because it has no direct analog in any other dimension.

2012, Andrzej Katunin, Fractals based on regular convex polytopes, page 56:

Shape of Nature comes from a unique, optimally symmetric geometric object: the sixth platonic element, the icositetrachoron (or 24-cell regular polytope), crystallized as a hyperdiamond network, mathematically and physically made of loops and connections.

2012, Raymond Aschheim, Spin foam with topologically encoded tetrad on trivalent spin networks:

(mathematics) A four-dimensional object with no regular three-dimensional analogue, constructed out of twenty-four octahedra.


The octagonal projection of the regular 24-cell {3,4,3} reveals that the 24 vertices of this 4-dimensional polytope can be distributed as 16 + 8: the 16 vertices of the 4-cube γ₄ = {4,3,3} and the 8 vertices of its dual, the 16-cell β₄ = {3,3,4}. This view of the 24-cell is less well-known than the dodecagonal projection, in which the β₄ appears as two squares of different sizes joined by 8 equilateral triangles.

1995, Harold Scott Macdonald Coxeter, “Paper Three: Two Aspects of the Regular 24-Cell in Four Dimensions”, in F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Iviċ Weiss, editors, Kaleidoscopes: Selected Writings of H.S.M. Coxeter, page 25:

2002, T. Robbin, Formian for art and mathematics, G. A. R. Parke, P. Disney (editors), Space Structures 5, Proceedings of the 5th International Conference on Space Structures, Volume 1, page 445,
One more example of a four dimensional tessellation is given using the 24-cell, see Fig 2.

The regular polytope with Schläfli symbol {3,4,3}, the so-called 24-cell, can be obtained from the 16-cell as follows. The vertices of the 24-cell are the midpoints of the 24 edges of the 16-cell.

(geometry) A four-dimensional polytope whose twenty-four bounding facets are octahedra and which has no three-dimensional analogue.

hyper-diamond

4-dimensional polytope with 24 octahedral facets


hecatonicosachoron

hecatonicosachoron (plural hecatonicosachorons or hecatonicosachora)

hecatonicosachorons (plural)
hecatonicosachora (plural)

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From hecaton- (“one hundred”) + icosa- (“twenty”) + -choron (“room”), from Ancient Greek ἑκατόν (hekatón, “one hundred”) εἴκοσι (eíkosi, “twenty”) and χῶρος (khôros, “room”).

uder

The pentachoron, the hecatonicosachoron, and the hexacosichoron all have dichoral angles which are not divisors of 2π, and consequently, they are not space-filling.

The hecatonicosachoron (dodecaplex or 120-cell) and the hexacosichoron (tet- raplex or 600-cell) are also unique polychora, which have no analogs in higher dimensions.

2012, Andrzej Katunin, Fractals based on regular convex polytopes, page 57:

The pentachoron, hypercube, orthoplex, hecatonicosachoron, and hexacosichoron are analogues of the ordinary Platonic solids.

2012, Richard Elwes, How to Solve the Da Vinci Code:

(geometry) A four-dimensional object analogous to a dodecahedron, constructed out of one hundred and twenty dodecahedra, arranged 4 to a vertex.

正百二十胞体

せいひゃくにじゅうほうたい


The triacontagonal projection of the 120-cell was first drawn by W. A. Wythoff, but not in reproducible form.

1964, Canadian Mathematical Bulletin, page 394:

In dimension four there are ten star polytopes which together with the 120-cell and 600-cell form the family of pentagonal polytopes (those regular 4-polytopes possessing axes of fivefold symmetry); all twelve have the same symmetry group.

1987, Jeffrey Hurst Butler, Hyperplane sections of regular star polytopes, page 28:

In 1985, M. Davis [3] gave the first explicit geometric construction of a closed hyperbolic 4-manifold by gluing together the opposite sides of a regular hyperbolic 120-cell P with dihedral angle 72°.

2006, John G. Radcliffe, “The Geometry of Hyperbolic Manifolds of Dimension a least 4”, in András Prékopa, Emil Molnár, editors, Non-Euclidean Geometries: János Bolyai Memorial Volume, page 270:

(geometry) A four-dimensional polytope, analogous to a dodecahedron, whose 120 bounding facets are dodecahedra.

正百二十胞体

せいひゃくにじゅうほうたい

Davis 120-cell

4-dimensional manifold in hyperbolic geometry constructed from a hyperbolic 120-cell


1995, J. Bokowski, On the Construction of Equifacetted 3-Spheres, Neil L. White, Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curaçao Conference, page 303,
P. McMullen has pointed out to the author that he has constructed an interesting equifacetted 3-sphere in his Ph.D.Thesis by starting with the 600-cell.

The dual to the 120-cell is the 600-cell. The 600-cell is a 4-dimensional convex regular polytope with 600 facets. Its Schläfli symbol of the^([sic]) 600-cell is {3, 3, 5}. It is a 4-dimensional analogue of the icosahedron, constructed by the golden ratio on the edges of an octohedron.

2005, Lawrence B. Crowell, Quantum Fluctuations of Spacetime, volume 25, World Scientific Series in Contemporary Chemical Physics, page 329:

The other two figures are the 120-cell and 600-cell which form a dual pair as is clear from their Schläfli symbols being the reverse of each other.

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

(geometry) A four-dimensional polytope, analogous to an icosahedron, whose 600 bounding facets are tetrahedra.


apeirohedron

en-noun

apeirohedron (plural apeirohedrons or apeirohedra)

apeirohedrons (plural)
apeirohedra (plural)

/əˌpiːɹɵˈhiːdɹən/

/əˌpeɪ̯ɹɵˈhiːdɹən/

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There are exactly 12 regular apeirohedra that in some sense are reducible and have components that are regular figures of dimensions 1 and 2. These apeirohedra are blends of a planar regular apeirohedron, and a line segment { } or linear apeirogon {∞}. This explains why there are 12 = 6·2 blended (or non-pure) apeirohedra. For example, the blend of the standard square tessellation {4,4} and the infinite apeirogon {∞}, denoted {4,4}#{∞}, is an apeirohedron whose faces are helical apeirogons (over squares), rising above the squares of {4,4}, such that 4 meet at each vertex; the orthogonal projections of {4,4}#{∞} onto their component subspaces recover the original components, the square tessellation and the linear apeirogon.

2014, Daniel Pellicer with Egon Schulte, “Polygonal Complexes and Graphs for Crystallographic Groups”, in Robert Connelly, Asia Ivić Weiss, Walter Whiteley, editors, Rigidity and Symmetry, New York, N.Y.: Springer, →ISBN, page 331:

(mathematics, geometry) A polyhedron with an infinite number of faces.


antimedial

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(geometry) Being a triangle formed by three lines parallel to the sides of an original triangle ABC: the parallel to AB through C, the parallel to AC through B, and the parallel to BC through A.


coordinate axis

coordinate axis (plural coordinate axes)

coordinate axes (plural)
co-ordinate axis (alternative)

(mathematics, geometry) One of the lines in the coordinate system that represents one of the dimensions.


orthoptic

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(geometry) The isoptic whose given angle is a right angle.


isoptic

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(geometry) The set of points for which two tangents of a given curve meet at a given angle.


digit

From Middle English digit, from Latin digitus (“a fingerbreadth; a number”). Doublet of digitus and dedo.

bit

position in a sequence of numerals, symbol in a positional number system

(geometry, rare, obsolete) Synonym of degree: ¹⁄₃₆₀ of a circle.


hypercell

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(geometry) A four-dimensional face of a polytope.


nephroid

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From nephro- + -oid, i.e. "kidney-shaped".

(geometry) A plane curve resembling two intersecting ellipses


cubo-cubic

Cremona transformation
Cremona transformation

(geometry, of a transformation) Being a Cremona transformation such that the homaloids of the transformation and its inverse all have degree 3.


tacnode

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

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

(geometry) A point of a curve where two or more osculating circles to the curve at that point are tangent, so that two branches of the curve have ordinary tangency at the double point.


antipoint

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(geometry) One of a pair of points constructed from two foci of a curve.


Aronhold set

Named after Siegfried Heinrich Aronhold.

(geometry) One of the 288 sets of seven of the 28 bitangents of a quartic curve corresponding to the seven odd theta characteristics of a normal set.


azygetic

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azygetic triad; azygetic tetrad; azygetic set


bicuspidal

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bielliptic

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(geometry, of a curve) Being a branched double cover of an elliptic curve.


biflecnode

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bigenus

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(geometry) The second plurigenus P₂ of a surface.


binode

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(geometry) A double point of a surface whose tangent cone consists of two different planes.


biregular

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(geometry, of a map) Being a regular map with regular inverse.

(geometry, of two varieties) Having a biregular map from one to the other, i.e. isomorphic as abstract varieties.


biscribed

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a biscribed triangle

(geometry) Both circumscribed and inscribed, i.e. having vertices that lie on a curve and sides that are tangent to the curve.


bitangential

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(geometry) Meeting a curve at the tangency points of its bitangents.


Brianchon hexagon

en-noun

Brianchon hexagon (plural Brianchon hexagons)

Named after Charles Julien Brianchon (1783–1864).

(geometry) A nonplanar hexagon whose three diagonals meet.


Zeuthen-Segre invariant

Zeuthen-Segre invariant (plural Zeuthen-Segre invariants)

Introduced by Zeuthen (1871) and rediscovered by Corrado Segre (1896).

(geometry) A certain invariant of complex projective surfaces.


orbiform

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(geometry) A plane for which the distance between the points of contact of two nonequal parallel tangents at a circle is the same everywhere.


ellipsoid of revolution

ellipsoid of revolution (plural ellipsoids of revolution)

An ellipsoid of revolution may be either prolate or oblate.

(geometry) A spheroid; a solid of revolution generated by rotating an ellipse about one of its axes.


spherical excess

(geometry) The amount by which the sum of the three angles of a spherical triangle exceeds two right angles. The spherical excess is proportional to the area of the triangle.


logocyclic curve

logocyclic curve (plural logocyclic curves)


strophoid

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(geometry) The locus of a point that has a certain relationship with a given curve


googolgon

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(geometry) A polygon with a googol number of sides (virtually indistinguishable from a circle)


ultraparallel

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(geometry, of lines in hyperbolic geometry) Sharing a common perpendicular.


power center

power centers (plural)
power centre (alternative)


quadrisecant

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(geometry) A line that passes through four points of the curve.


homaloid

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(geometry) An element of a homaloidal system, in particular the image of a hyperplane under a Cremona transformation.


hypercusp

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(geometry) A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve with order r+1.


hyperflex

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(geometry) A point of a curve where the tangent line has contact of order at least 4.


hyperosculate

en-verb

hyperosculate (third-person singular simple present hyperosculates, present participle hyperosculating, simple past and past participle hyperosculated)

hyperosculates (present singular third-person)
hyperosculating (participle present)
hyperosculated (participle past)
hyperosculated (past)

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A hyperosculating point is one where the tangent space meets with order higher than normal.

(geometry) To osculate at multiple points.


flecnode

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(geometry) A double point that is also a point of inflexion of one branch.


fleflecnode

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(geometry) A double point that is also a point of inflexion of both branches.


equianharmonic

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(geometry) A set of four points whose cross ratio (or anharmonic ratio) is a cube root of 1.


neusis

(geometry) A line segment of prescribed length, fitted between two given curves, part of a line that passes through a given point.


tendril perversion

en-noun

tendril perversion (countable and uncountable, plural tendril perversions)

(geometry) A reversal of handedness in helical curves.


visual angle

(geometry) The angle formed by two rays of light, or two straight lines drawn from the extreme points of an object to the center of the eye.


winding number

mathematical analysis
mathematical analysis

(geometry, topology, mathematical analysis) The total number of times a given curve travels counterclockwise around a specified point.

indeks

the number of times a curve travels counterclockwise around a point


Bravais lattice

Bravais lattice (plural Bravais lattices)

(geometry, crystallography) An infinite array of discrete points generated by a set of discrete translation operations mathbf R=n_1 mathbf a₁+n_2 mathbf a₂+n_3 mathbf a₃, where nᵢ are any integers and aᵢ are known as the primitive vectors which lie in different directions and span the lattice.


geometrography

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(geometry) The study of geometrical constructions.


lensoidal

en-adj

lensoidal (comparative more lensoidal, superlative most lensoidal)

more lensoidal (comparative)
most lensoidal (superlative)

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(geometry) Of a lenticular shaped geometry.


excenter

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(US, geometry) The center of an excircle.


exradius

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Coxeter-Dynkin diagram

en-noun

Coxeter-Dynkin diagram (plural Coxeter-Dynkin diagrams)

After mathematicians H. S. M. Coxeter and Eugene Dynkin.

A Coxeter-Dynkin diagram encodes the information in a Coxeter matrix, which in turn encodes the presentation of a Coxeter group.

Vertices of a Coxeter-Dynkin diagram represent generators of the corresponding Coxeter group. The generators (algebraic) in turn correspond to the reflecting hyperplanes (geometric). A pair of vertices which are not linked by an edge correspond to a pair of commuting generators. An unnumbered edge between a pair of vertices means that the braid relation between the corresponding generators has length three (e.g., aba = bab if the generators are a and b). An edge numbered ≥4 means that the braid relation between the corresponding generators has a length equal to that number. For example, if the edge is numbered 4 then the braid relation is cdcd = dcdc if the generators are c and d. If a set of edges form a cycle then the Coxeter group can be shown to be infinite. If a tree in a Coxeter-Dynkin diagram has more than one numbered edge then the Coxeter group can be shown to be infinite. There are a few more such rules, which ensure that finite Coxeter groups have Coxeter-Dynkin diagrams with relatively simple shapes.

Let 𝒬 be an indecomposable root lattice and let Γ denote the Coxeter-Dynkin diagram of the underlying root system Δ.

1995 June, R. V. Moody, J. Patera, “Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices”, in Canadian Journal of Mathematics, page 597:

There is a deep and only partially understood connection between the classification and structure of singularities and the Coxeter-Dynkin diagrams introduced by H. S .M. Coxeter for classification of reflection-generated groups and by E. B. Dynkin for classification of semisimple Lie algebras.

2000, Andrei Gabrielov, “Coxeter-Dynkin diagrams and singularities”, in Evgeniĭ Borisovich Dynkin, A. A. Yushkevich, Gary M. Seitz, A. L. Onishchik, editors, Selected Papers of E. B. Dynkin with Commentary, page 367:

For 3 ≤ n ≤ 5, we will use Eₙ to denote the Coxeter-Dynkin diagrams of types A₁ + A₂(N = 3), A₄(N = 4) and D₅(N = 5).

2012, Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View, page 363:

(geometry, algebra) A graph with numerically labelled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

Coxeter-Dynkin diagram
Eugene Dynkin
Harold Scott MacDonald Coxeter


demihypercube

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(geometry) Any of a class of n-polytopes constructed from alternation of an n-hypercube.


bicylinder

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(geometry) A solid formed by two intersecting cylinders.


tricylinder

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(geometry) A solid formed by three intersecting cylinders.


Steinmetz solid

Steinmetz solid (plural Steinmetz solids)

Named after Charles Proteus Steinmetz, though these solids were known long before Steinmetz studied them.

(geometry) The solid body generated by the intersection of two or three cylinders of equal radius at right angles.


golygon

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(geometry) A serial isogon of 90 degrees; any polygon with all right angles, whose sides are consecutive integer lengths.


acute angle

(geometry) An angle greater than 0 degrees and less than 90 degrees.

زَاوِيَة حَادَّة

angle measuring greater than zero degrees and less than ninety degrees


obtuse angle

(mathematics, geometry) An angle that is greater than 90 degrees and less than 180 degrees.


reflex angle

(geometry) An angle greater than 180 degrees but less than 360 degrees.


parallel postulate

parallel postulate (plural parallel postulates)

From the reference to parallel lines in the definition as formulated below, following Scottish mathematician John Playfair; this wording leads to a convenient basic categorization of Euclidean and non-Euclidean geometries. The original formulation in Euclid's Elements makes no mention of parallels.

Before we examine the particulars of the parallel postulates of Hyperbolic and Elliptical geometries, we must see their logic.

1962, Mary Irene Solon, The Parallel Postulates of Non-Euclidean Geometry: The Pentagon: A Mathematics Magazine for Students, Volumes 22-25, page 28:

The earliest known attempt to prove the parallel postulates from the other axioms was by Ptolemy in about 150 A.D.

1969, John B. Fraleigh, Mainstreams of Mathematics, Addison-Wesley, page 95:

Euclid's parallel postulate may not have seemed so important when you studied geometry in high school, since it is used only once in order to derive the basic result 1 on alternate interior angles, which is then constantly used to derive further results.

1989, Walter Prenowitz, Meyer Jordan, Basic Concepts of Geometry, Ardsley House Publishers, published 1965, page 28:

In order to understand this section, it is vital to keep in mind an important fact: The "proof" of the parallel postulate presented here is not correct. In fact, there can be no proof of the parallel postulate that relies only on the other axioms and postulates of Euclid.

1998, David A. Singer, Geometry: Plane and Fancy, Springer, page 16:

After enduring twenty centuries of criticism, Euclid's theory of parallels was fully vindicated in 1868 when Eugenio Beltrami proved the independence of Euclid's parallel postulate by constructing a Euclidean model of the hyperbolic plane.

2006, John Ratcliffe, Foundations of Hyperbolic Manifolds, Springer, page 7:

(geometry) An axiom in Euclidean geometry: given a straight line L and a point p not on L, there exists exactly one straight line parallel to L that passes through p; a variant of this axiom, such that the number of lines parallel to L that pass through p may be zero or more than one.

American Mathematical Society
Euclid's Elements
John M. Lee
John Playfair
parallel postulate


coaxal

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sphenocorona

Blend of sphenoid + corona or from spheno- + corona. Coined by Norman W. Johnson in 1969.

(geometry) A polyhedron, one of the Johnson solids, whose faces are composed of 12 equilateral triangles and 2 squares.


sphenomegacorona

From spheno- + mega- + corona. Coined by Norman W. Johnson in 1969.

(geometry) A polyhedron, one of the Johnson solids, whose faces are composed of 16 triangles and 2 squares.


augmented sphenocorona

(geometry) A polyhedron, one of the Johnson solids, whose faces are composed of 16 equilateral triangles and one square.


triangular hebesphenorotunda

(geometry) A polyhedron, one of the Johnson solids, whose faces are composed of 13 triangles, 3 squares, 3 pentagons, and 1 hexagon.


Eckardt point

(geometry) A point of intersection of three lines on a cubic surface.


equiaffine

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(geometry) Being an affine transformation that preserves area.


equiaffinity

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(geometry) An affine transformation that preserves area.


evectant

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(geometry) A contravariant constructed from an invariant by acting on it with a differential operator called an evector.


evector

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(geometry) A differential operator that allows a contravariant to be constructed from an invariant.


Schläfli double six

Schläfli double six (plural Schläfli double sixes)

Schläfli double sixes (plural)
Schläfli's double six (alternative)

(geometry) A configuration of 30 points and 12 lines. The lines can be partitioned into two subsets of six: each line is disjoint from (skew with) the lines in its own subset of six lines, and intersects all but one of the lines in the other subset of six lines. Each of the 12 lines of the configuration contains five intersection points, and each of these 30 intersection points belongs to exactly two lines, one from each subset.


Desargues configuration

Desargues configuration (plural Desargues configurations)

(geometry) A configuration of ten points and ten lines, with three points per line and three lines per point, whose existence is proven by Desargues' theorem.


perspectivity

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(geometry) The formation of an image in a picture plane of a scene viewed from a fixed point.


desmic

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(geometry, of polyhedra) Related such that each edge of one cuts a pair of opposite edges of the other.


orthoplex

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Coined in 1991 by John Horton Conway and Neil Sloane. Blend of orthant + complex, since it has one facet for each orthant.

We made use of another way of considering the 120 cells, starting with the eight at the vertices of an orthoplex, that is, in the cells of a hypercube.

1990, Mathematica Journal, volumes 1-2, page 84:

It is remarkable that the four-dimensional orthoplex is the same polytope as the four-dimensional hemicube.

1991, J. H. Conway, N. J. A. Sloane, “The Cell Structures of Certain Lattices”, in Peter Hilton, Friedrich Hirzebruch, Reinhold Remmert, editors, Miscellanea Mathematica, page 90:

The combinatorics of this case apply to all members of the Gosset series; in every case, their cells are simplexes and orthoplexes, the latter appearing with only half symmetry.

2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, page 412:

(geometry) A convex polytope analogous to an octahedron (3 dimensions) or 16-cell (4 dimensions).

Cross-polytope
John Horton Conway
Neil Sloane


CPCTC

(geometry) Initialism of corresponding parts of congruent triangles are congruent, a theorem regarding congruent triangles, stating that if two or more triangles are proven congruent by any method, then all of their corresponding angles and sides are congruent as well.


CPCFC

(geometry) Initialism of corresponding parts of congruent forms are congruent.


tridiminished

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(geometry) Having had three cupolae removed.


bidiminished

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(geometry) Having had two cupolae removed.


triaugmented

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(geometry) Having a square pyramid attached to each of three equatorial faces.


deltahedron

deltahedron (plural deltahedrons or deltahedra)

deltahedrons (plural)
deltahedra (plural)

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From delta (Greek letter) + -hedron referring to the triangular shape of the letter.

There are infinitely many deltahedra, but only eight that are convex.

(geometry) A polyhedron whose faces are all equilateral triangles.


straight angle

(mathematics, geometry) An angle that is equal to 180° (pi radians or two right angles).


Pi Day

pi + day, from the date March 14th in the month/day format (that is, 3/14), 3, 1 and 4 being the first three significant digits of the mathematical constant π.

Did you know that March 14 is Pi Day (no, π is not misspelled). Pi Day is celebrated on 3/14 or 3.14! To commemorate this remarkable ratio, we have a few circular activities. The first is the Pizza Survey. What are the students' favorite toppings on pizza? What do you like best?

1997, Louise Bock, Susan Guengerich, Hope Martin, “March Celebrations”, in Multicultural Math Fun: Holidays around the Year, Portland, Me.: J. Weston Walch Publisher, →ISBN, page 83:

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

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

Of all known mathematical constants, however, pi continues to attract the most attention. […] The Exploratorium in San Francisco continues to celebrate Pi Day on March 14 each year, starting at 1:59 pm.

2002, Ivan Peterson, “A Passion for Pi”, in Mathematical Treks: From Surreal Numbers to Magic Circles (Spectrum Series; a Science News Book), [Washington, D.C.]: Mathematical Association of America, →ISBN, page 17:

Whereas Pi can be approximated as 3.14, and thus March 14, 2009, is an appropriate day for "National Pi Day": Now, therefore, be it / Resolved, That the [United States] House of Representatives— / (1) supports the designation of a "Pi Day" and its celebration around the world; […]

2009 March 12, H. Res. 224 Supporting the Designation of Pi Day, and for Other Purposes, [Washington, D.C.]: United States Government Publishing Office, archived from the original on 2016-12-28, page 2:

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

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

March 14, or 3/14, is recognized by math geeks as Pi day. That happens every year. But as UBS’s Art Cashin notes, next year’s Pi day will be a once-in-a-century Pi day. “At 9:26:53 a.m., they can write 3/14/15; 9:26:53, which you should also recall are the first ten digits of Pi in perfect order,” said Cashin.

2015 March 14, Sam Ro, “March 14, 2015 will be a once-in-a-century thrill for math geeks”, in Business Insider, archived from the original on 2016-03-04:

Pi lovers like to celebrate Pi Day every year. In the US, they write the date with the month first, then the day and year. So in 2015, 14th March was 3 14 15, the first five digits of π. So that was a really special Pi Day when, of course, they sang their special song, Pi, Pi, Mathematical Pi, to the tune of that great song everyone danced to in the 1970s, 'Bye-bye, Miss American Pie' by Don McLean.

2016, Liz Strachan, “A Little More Pi”, in Easy as Pi: Maths Made Simple, London: Robinson, →ISBN:

March 14th, an annual celebration of the mathematical constant π (pi).


octadecagon

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A regular octadecagon cannot be constructed using a compass and straightedge.

(geometry, rare) A polygon with 18 sides and 18 vertices.


omega point

To begin, let AB be a given segment, and l and m be sensed parallel lines with shared omega point Ω.

2018, Maureen T. Carroll, Elyn Rykken, Geometry: The Line and the Circle, page 319:

(geometry) A well-defined point outside the hyperbolic plane.


octogram

octograms (plural)
octagram (alternative)

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(geometry) A regular eight-pointed star shape.


nonagram

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(geometry) A regular nine-pointed star shape.


pseudospherical

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quadrilinear

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(geometry) Having, or bounded by, four lines.


antiprismatic

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Hadwiger's theorem

(geometry) A theorem asserting that every valuation on convex bodies in Rⁿ that is continuous and invariant under rigid motions of Rⁿ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).


hypertorus

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(geometry, topology) An N-dimensional equivalent of a torus, where N is a positive integer greater than 3.


Fano plane

Named after Gino Fano (1871–1952), an Italian mathematician.

(geometry) The finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point.


Catalan surface

Catalan surface (plural Catalan surfaces)

Named after the Belgian mathematician Eugène Charles Catalan.

(geometry) A ruled surface whose rulings are all parallel to a fixed plane.


Brocard triangle

Brocard triangle (plural Brocard triangles)

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

(geometry) The triangle formed by joining the points of intersection of the lines from each vertex of a triangle to its Brocard points.


Brocard point

(geometry) The point inside a triangle at the intersection of the lines through each vertex such that the angle between the line and the side of the triangle when moving clockwise is the same for each vertex.


nonsqueezing

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

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

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


parallelohedron

parallelohedron (plural parallelohedrons or parallelohedra)

parallelohedrons (plural)
parallelohedra (plural)

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(geometry) A polyhedron that can tessellate three-dimensional spaces with face-to-face contacts via translations.


Hilbert curve

First described by the German mathematician David Hilbert in 1891.

(geometry) A continuous fractal space-filling curve.


hyperball

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

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

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


hyperbolic geometry

hyperbolic geometry (countable and uncountable, plural hyperbolic geometries)

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

parallel postulate
parallel postulate#English

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

(geometry) A non-Euclidean geometry, that features the hyperbola as geodesic, and has constant negative curvature


tesseractic

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


trisector

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(geometry) A line or curve that trisects something.


spinode

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starlikeness

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(geometry) The property of being starlike.


rhombitrihexagonal

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


isotoxal

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

edge-symmetric

having symmetries that act transitively on its edges


acylindrical

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scalebound

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

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


sextantal

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apeirogonal

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(geometry) Having the form of an apeirogon.


circumscribed circle

circumscribed circle (plural circumscribed circles)


inscribed circle

inscribed circle (plural inscribed circles)


point of tangency

point of tangency (plural points of tangency)

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

切點 /切点

the point where a tangent line and its curve meet


Coxeter group

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

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

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

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

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

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

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

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

Coxeter group
Harold Scott MacDonald Coxeter


vertex figure

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

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

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

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

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

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


oloid

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

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

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

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


spirallikeness

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

(geometry) The property of being spirallike.


rhombic dodecahedron

rhombic dodecahedron (plural rhombic dodecahedra or rhombic dodecahedrons)

rhombic dodecahedra (plural)
rhombic dodecahedrons (plural)
rhombicdodecahedron (alternative)

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

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

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

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

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

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

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


Schläfli symbol

Schläfli symbol (plural Schläfli symbols)

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

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

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

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

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

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

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

Schläflin symboli

notation encoding certain properties of a regular polytope or tessellation


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

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

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

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

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

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


tetrakis hexahedron

tetrakis hexahedron (plural tetrakis hexahedra or tetrakis hexahedrons)

tetrakis hexahedra (plural)
tetrakis hexahedrons (plural)
tetrakishexahedron (alternative)
tetrakis-hexahedron (alternative)

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

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

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

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

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

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

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

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

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

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

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

tetrakis-heksaedri

polyhedron with 24 triangular faces and 36 edges


Schlegel diagram

Schlegel diagram (plural Schlegel diagrams)

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

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

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

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

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

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

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

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


angular defect

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

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

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

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

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

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

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

angular deficiency

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

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

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

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

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

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


symmetry group

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

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

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

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

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

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

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

full symmetry group

emphasises the inclusion of orientation-reversing transformations