ARTIKELDIGITAL.COM

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries...

Isogonal, a mathematical term meaning "having similar angles", may refer to: Isogonal figure or polygon, polyhedron, polytope or tiling Isogonal trajectory...

equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For...

In geometry, the isogonal conjugate of a point P with respect to a triangle △ABC is constructed by reflecting the lines PA, PB, PC about the angle bisectors...

A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value,...

not lie on any of the extended sides BC, CA, AB, and let P −1 denote the isogonal conjugate of P. The pedal triangle of P is homothetic to the antipedal...

Unicode glyph ۞  at U+06DE. Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices...

e. isogonal. The Catalan solids, the Platonic Solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean...

called isogonal lines, will also have interesting properties. For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines...

tangent to this hyperbola at I {\displaystyle I} . The hyperbola is the isogonal conjugate of O I {\displaystyle OI} , the line joining the circumcenter...

truncations of the regular pentadecagon and pentadecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices...

intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory. Generally, one assumes that the pencil of curves is given implicitly...

omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon...

in the internal angle, α. {∞α} is the dual polygon of an isogonal skew apeirogon. An isogonal skew apeirogon alternates two types of edges with various...

symmetry is labeled a1. The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges...

Newton–Gauss line of the complete quadrilateral ABCDEF and the line EF are isogonal lines of ∠BEC, that is, each line is a reflection of the other about the...

If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits...

equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal 4 n {\displaystyle 4n} -gons are centrally symmetric...

rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon...

directed edges. The highest symmetry irregular tetradecagons are d14, an isogonal tetradecagon constructed by seven mirrors which can alternate long and...

polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows...

Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices...

B, C respectively. The isogonal conjugate of P is sometimes denoted by P*. The isogonal conjugate of P* is P. A self-isogonal cubic is a triangle cubic...

equiangular: all its corner angles are equal (each of 90 degrees). It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit. It...

Isogonal truncations of pentagon and pentagram Quasiregular Isogonal Quasiregular Double covering t{5} = {10} t{5/4} = {10/4} = 2{5/2} t{5/3} = {10/3}...

circumcircle. Tangential: all sides are tangent to an inscribed circle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The...

gyration orders. The most common high symmetry hexadecagons are d16, an isogonal hexadecagon constructed by eight mirrors can alternate long and short edges...

Coxeter–Dynkin diagrams Symmetry group Dihedral (D7) Internal angle (degrees) ≈77.143° Properties star, cyclic, equilateral, isogonal, isotoxal Dual polygon self...

tilings are isogonal and monohedral. A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement...

rotation, translation, reflection, glide reflection, group, transitivity, k-isogonal tiling (vertex-transitive), §1.4 symmetry element, isomorphism, affine...

the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are uniform polyhedra. They form one of the two infinite series...

isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive). Quasiregular polyhedra are isogonal and isotoxal, but not isohedral;...

circle. If P {\displaystyle P} does not lie on the circumcircle then its isogonal conjugate Q {\displaystyle Q} yields the same pedal circle, that is the...

arrangement: 2{12}, 3{8}, 4{6}, 6{4}, 8{3}, 3{8/3}, and 2{12/5}. There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon...

dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equally spaced vertices can be constructed...

polygon, {p/q}, is a special case of a spirolateral, 1180(1−2q/p)°. An isogonal polygon, is a special case spirolateral, 2180(1−2/p)° or 2180(1−2q/p)°...

triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates. Let X = u ( a , b , c ) : v ( a , b , c ) : w ( a , b , c )...

as directed edges. The highest symmetry irregular decagons are d10, an isogonal decagon constructed by five mirrors which can alternate long and short...

letter and group order. r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges...

result: For any point P in the plane of △ABC, the isogonals of the lines AP, BP, CP are concurrent, the isogonal of AP (respectively BP, CP) being the reflection...

conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.) Isogonal conjugate Triangle center Robert Lachlan, An Elementary...

Examples of isogonal tilings with isotoxal simple stars Isotoxal simple n-pointed stars "Triangular" stars (n = 3) "Square" stars (n = 4) "Hexagonal" stars...

Coxeter-Dynkin diagram 4-faces 242 Cells 2400 Faces 6000 Edges 5760 Vertices 1920 Vertex figure Coxeter group B5 [4,3,3,3] Properties convex, isogonal...

{3} 90 {4} Edges 120 Vertices 30 Vertex figure Tetrahedral antiprism Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal, isotoxal...

as directed edges. The highest symmetry irregular icosagons are d20, an isogonal icosagon constructed by ten mirrors which can alternate long and short...

Rectangular cuboid Type Prism Plesiohedron Faces 6 rectangles Edges 12 Vertices 8 Properties convex, zonohedron, isogonal...

a noble polyhedron, because it is both isohedral (face-transitive) and isogonal (vertex-transitive). Coxeter et al. (1999), p. 30–31; Wenninger (1971)...

snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon...

Tetrahedron Petrie polygon octagon Coxeter group B4, [3,3,4] Dual 16-cell Properties convex, isogonal, isotoxal, isohedral, Hanner polytope Uniform index 10...

A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). They were first studied in any depth by Edmund...

Deeper truncations of the regular enneagon and enneagrams can produce isogonal (vertex-transitive) intermediate octadecagram forms with equally spaced...

compound star figures with the same vertex configuration. There are also isogonal triacontagrams constructed as deeper truncations of the regular pentadecagon...

regular faces and has a symmetry of mapping any two vertices known as isogonal. The dihedral angle between two adjacent square faces is the internal angle...

JSTOR 3619822. S2CID 195047512. See p. 247. Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18 (1): 13–52. doi:10...

figures. Vertex figures are especially significant for uniforms and other isogonal (vertex-transitive) polytopes because one vertex figure can define the...

diagrams or Symmetry group Dihedral (D6) Internal angle (degrees) 60° Properties star, compound, cyclic, equilateral, isogonal, isotoxal Dual polygon self...

the associated angle bisectors) of a triangle. In other words, it is the isogonal conjugate of the centroid of a triangle. Ross Honsberger called its existence...

dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries...

tilings is the same as the uniform tilings. Because the uniform tilings are isogonal, the duals are isohedral. The k-uniform tilings with valence-5 vertices...

symbol {}×{}×···×{} = {}n Coxeter diagram ··· Symmetry group [2n−1], order 2n Dual polyhedron Rectangular n-fusil Properties convex, zonohedron, isogonal...

Symmetry group Dihedral (D11), order 2×11 Internal angle (degrees) ≈147.273° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

Symmetry group Dihedral (D7), order 2×7 Internal angle (degrees) ≈128.571° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

Isogonal polytope with regular facets...

Symmetry group Dihedral (D1000), order 2×1000 Internal angle (degrees) 179.64° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

isotoxal (or edge-transitive, which involves the edge's polyhedra), and isogonal (or vertex-transitive, which involves the polyhedra's vertices). For example...

graph, i.e. a graph representing the edge adjacencies of another graph Isogonal conjugate, in geometry Conjugate gradient method, an algorithm for the...

self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal)...

known as the triangle's Kosnita point (named by Rigby in 1997). It is the isogonal conjugate of the nine-point center. It is triangle center X(54) in Clark...

second Brocard point of △ACB. The two Brocard points of a triangle △ABC are isogonal conjugates of each other. The most elegant construction of the Brocard...

group Dihedral (D10000), order 2×10000 Internal angle (degrees) 179.964° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

y,z,a,b,c)=0} . The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of △ABC. A construction...

Truncated icosahedron (regular faces) Convex hull (isogonal hexagons) Small snub icosicosidodecahedron...

was found by Stefanovic (2010). We can infer the proof from existence of isogonal conjugate too. If we are to show that X = AB ∩ DE, Y = BC ∩ EF, Z = CD...

The Thomson cubic (black curve) contains the point X whose isogonal conjugate (X') is on the line X(2)-X....

x y = 0 , {\displaystyle uyz+vzx+wxy=0,} for some point u : v : w. The isogonal conjugate of each point X on the circumconic, other than A, B, C, is a...

figure Tetrahedral prism Coxeter group B5, [4,33], order 3840 Dual Base point (0,1,1,1,1,1)√2 Circumradius sqrt(2) = 1.414214 Properties convex, isogonal...

polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows...

(Saules meita). Auseklis is also the name of the eight-pointed star (an isogonal octagram). It is also known as one of the crosses of Lietuvēns (a malevolent...

An isogon may refer to: Isogonal figure - a polygon or polyhedron with all of its vertices equivalent under the symmetries of the figure. A type of contour...

Symmetry group Dihedral (D257), order 2×257 Internal angle (degrees) 178.599222° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

diagrams Symmetry group Dihedral (D9), order 2×9 Internal angle (degrees) 140° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

Dihedral (D1000000), order 2×1000000 Internal angle (degrees) 179.99964° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

257–263. doi:10.5951/MT.45.4.0257. JSTOR 27954012. Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18: 13–52. doi:10.1007/PL00009307...

of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3-fold symmetry). E. L. Elte identified it in 1912 as a semiregular...

respectively denote the Boolean variables (A < 120°), (B < 120°), (C < 120°). The isogonal conjugate of X(13) is the first isodynamic point, X(15): sin ⁡ ( A + π...

from the square tiling. For 16-gons (n=4), the gaps can be filled with isogonal octagons and isosceles triangles. Critchlow, pp. 62–67 Grünbaum and Shephard...

group Dihedral (D65537), order 2×65537 Internal angle (degrees) ≈179.994 507° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

The final stellation of the icosahedron has 2-isogonal enneagram faces. It is a 9/4 wound star polyhedron, but the vertices are not equally spaced. The...

different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p2 rectangular trapezoprisms (a cube with...

B5, [4,33], order 3840 Dual 5-orthoplex Base point (1,1,1,1,1,1) Circumradius sqrt(5)/2 = 1.118034 Properties convex, isogonal regular, Hanner polytope...

Symmetry group Dihedral (D17), order 2×17 Internal angle (degrees) ≈158.82° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

{\displaystyle {\tfrac {a^{2}}{x}}+{\tfrac {b^{2}}{y}}+{\tfrac {c^{2}}{z}}=0.} The isogonal conjugate of the circumcircle is the line at infinity, given in trilinear...

Association of America, p. 85, ISBN 978-0-88385-358-0. Grünbaum, Branko (1997), "Isogonal Prismatoids", Discrete & Computational Geometry, 18: 13–52, doi:10.1007/PL00009307...

analytical solution to the general quintic equation. The regular icosahedron is isogonal, isohedral, and isotoxal: any two vertices, two faces, and two edges of...

diameter d OC = s csc ⁡ ( π n ) {\displaystyle d_{\text{OC}}=s\csc \left({\frac {\pi }{n}}\right)} Properties Convex, cyclic, equilateral, isogonal, isotoxal...

the United Kingdom Noble polyhedron, a polyhedron which is isohedral and isogonal North of Boston Library Exchange, or NOBLE, a cooperative effort of 28...

hyperbola, X(190) = Yff parabolic point, X(514) = Isogonal conjugate of X(101), X(523) = Isogonal conjugate of X(110), X(647) = Crossdifference of X(2)...

and excircles of a triangle Inellipse Integer triangle Isodynamic point Isogonal conjugate Isoperimetric point Isosceles triangle Isosceles triangle theorem...

combinatorially equivalent polyhedra with equal corresponding dihedral angles are isogonal. This conjecture has been proved for some special cases. The next result...

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex...

Encyclopedia of Triangle Centers. The image of the Kiepert hyperbola under the isogonal transformation is the Brocard axis of triangle A B C {\displaystyle ABC}...

edge and two short edges. Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex...

In geometry, the truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex...

The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely...

figure (Elongated equilateral-triangular antiprism) Symmetry group Aut(A4), [[3,3,3]], order 240 Properties convex, isogonal isotoxal Uniform index 4 5 6...

If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits...

isodynamic point is the one with minimum area. The isodynamic points are the isogonal conjugates of the two Fermat points of triangle A B C , {\displaystyle...

In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on...

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and...

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}. This regular tiling can also be...

Isogonal honeycomb of uniform polytope facets...

In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}. The symmetry is...

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}. John H. Conway, Heidi...

diagrams Symmetry group Dihedral (D5), order 2×5 Internal angle (degrees) 108° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

A6, [35], order 5040 Bowers name and (acronym) Rectified heptapeton (ril) Vertex figure 5-cell prism Circumradius 0.845154 Properties convex, isogonal...

5-cell Coxeter group A5, [34], order 720 Dual self-dual Base point (0,0,0,0,0,1) Circumradius 0.645497 Properties convex, isogonal regular, self-dual...

orbit, such that the appearance remains unchanged; these are isohedral, isogonal, and isotoxal respectively. Hence, it is considered a regular polyhedron...

Encyclopedia of Triangle Centers, the outer Vecten point is denoted by X(485). Isogonal conjugate of outer Vecten point is Kenmotu point [ja]. Let △ABC be any...

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons...

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling...

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where...

{BH}}{\overline {BE}}}+{\frac {\overline {CH}}{\overline {CF}}}=2.} The isogonal conjugate of the orthocenter is the circumcenter of the triangle. The isotomic...

Isogonal polyhedron with regular faces...

inverse with respect to the circumcenter of T {\displaystyle T} of the isogonal conjugate or the nine-point center of T {\displaystyle T} . The common...

xz-plane and the yz-plane. The sphenocorona is also the vertex figure of the isogonal n-gonal double antiprismoid where n is an odd number greater than one,...

irregular triangles and pentagons. A related isogonal tiling that combines pairs of triangles into rhombi A 2-isogonal tiling can be made by combining 2 squares...

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex...

Vertex figure Triangular prism Symmetry group A4, [3,3,3], order 120 Petrie polygon Pentagon Properties convex, isogonal, isotoxal Uniform index 1 2 3...

the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some...

In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}. Drawn in chiral pairs, with...

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two...

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}. This tiling represents a hyperbolic...

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal...

In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}. John H. Conway, Heidi...

In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There...

}}\triangle BCG)={\tfrac {1}{3}}({\text{Area of }}\triangle ABC).} The isogonal conjugate of a triangle's centroid is its symmedian point. Any of the three...

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal,...

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction...

the line XX −1 is tangent to the circumcircle, where X −1 denotes the isogonal conjugate of X. This equilateral triangle, called the circumtangential...

with its vertices on the sphere represents an ideal polyhedron. Every isogonal convex polyhedron (one with symmetries taking every vertex to every other...

In geometry, an apeirogonal hosohedron or infinite hosohedron is a tiling of the plane consisting of two vertices at infinity. It may be considered an...

uniform polyhedron), but is not a uniform polyhedron (due to not being isogonal), and has a smaller symmetry group. It can be obtained from the great rhombicuboctahedron...

Vertex figure tetrahedron Petrie polygon 30-gon Coxeter group H4, [3,3,5] Dual 600-cell Properties convex, isogonal, isotoxal, isohedral Uniform index 32...

locus of points X {\displaystyle X} such that X {\displaystyle X} , the isogonal conjugate of X {\displaystyle X} , and the de Longchamps point are collinear...

the Nagel point is the incenter of the anticomplementary triangle. The isogonal conjugate of the Nagel point is the point of concurrency of the lines joining...

In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex...

In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex...

triangle's opposite vertex, are concurrent at a point called the first isogonal center. In the case in which the original triangle has no angle greater...

[[3,4,3]]+ isogonal 4-polytope # Name Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3-0 (48) Pos. 2-1...

of a point P on the Euler line is perpendicular to the line through the isogonal conjugate and the anticomplement of P. The orthotransversal of the nine-point...

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular...

In geometry, the rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{4,5}. The dual is called the...

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and...

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon...

vertices in parallel planes and are therefore prismatoids. Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a...

In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams. This tiling is a regular star-tiling, and has...

lectures. He is well remembered by the Jerabek hyperbola, the locus of the isogonal conjugate of a point that traverses the Euler line of a triangle. He was...

In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3...

group B4, [3,3,4], order 384 D4, order 192 Dual Tesseract Properties convex, isogonal, isotoxal, isohedral, regular, Hanner polytope Uniform index 12...

In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal...

dodecahedron, as the family of Platonic solids, is a regular polyhedron. It is isogonal, isohedral, and isotoxal: any two vertices, two faces, and two edges of...

https://mathworld.wolfram.com/OuterNapoleonTriangle.html Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18: 13–52. doi:10.1007/PL00009307...

In geometry, the snub tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,4}. Drawn in chiral pairs, with...

Petrie polygon pentagon Coxeter group A4, [3,3,3] Dual Self-dual Properties convex, isogonal, isotoxal, isohedral, projectively unique Uniform index 1...

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one square, one hexagon, and one tetradecagon...

or stephanoid is a toroidal polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting...

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}. The dual of this...

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}. This tiling is topologically related...

Symmetry group Dihedral (D13), order 2×13 Internal angle (degrees) ≈152.308° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

162 Cells 1440 Faces 4160 Edges 4800 Vertices 1920 Vertex figure Irregular 5-cell Coxeter groups B5 [4,3,3,3] D5 [32,1,1] Properties convex, isogonal...

k-dual-uniform or k-isohedral; if there are t orbits of dual vertices, as t-isogonal; if there are e orbits of edges, as e-isotoxal. k-dual-uniform tilings...

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one...

{}x{3} 30 t02{3,3} Faces 420 240 {3} 180 {4} Edges 360 Vertices 90 Vertex figure Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal...

17 October 2025. See X(351) = Center of the Parry circle and X(512) = Isogonal conjugate of X(99). Weisstein, Eric W. "Steiner Circumellipse". MathWorld--A...

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each...

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular...

some representations of quartz crystals. Uninodality corresponds with isogonality in geometry and vertex-transitivity in graph theory, and produces examples...

In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,5}, constructed from five pentagons...

In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)}...

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}. This tiling represents a hyperbolic...

Symmetry group Dihedral (D23), order 2×23 Internal angle (degrees) ≈164.348° Properties Convex, cyclic, equilateral, isogonal, isotoxal Dual polygon Self...

4200 {6} 1260 {4} Edges 15120 Vertices 5040 Vertex figure irregular 5-simplex Coxeter group A6, [[35]], order 10080 Properties convex, isogonal, zonotope...

In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,5}. John H. Conway, Heidi...

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}. A half symmetry...

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}. There are for uniform constructions...

{3}×{6} 15 { }×r{3,3} 6 rr{3,3,3} Cells 315 Faces 720 Edges 630 Vertices 180 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex, isogonal...

In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling...

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and one heptagon on each vertex...

icosahedron Petrie polygon 30-gon Coxeter group H4, [3,3,5], order 14400 Dual 120-cell Properties convex, isogonal, isotoxal, isohedral Uniform index 35...

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}. This tiling can also...

In geometry, the snub infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}. John H. Conway, Heidi...

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}. There...

dual polytope Disphenoidal 288-cell Symmetry group Aut(F4), [[3,4,3]], order 2304 Properties convex, isogonal, isotoxal, isochoric Uniform index 26 27 28...

In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}. The dual tiling, called...

triangle, a triangle center associated with Kosnita's theorem, is the isogonal conjugate of the nine-point center. Trilinear coordinates for the nine-point...

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}. This tiling can...

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be...

F4, [3,4,3], order 1152 B4, [4,3,3], order 384 D4, [31,1,1], order 192 Dual Self-dual Properties convex, isogonal, isotoxal, isohedral Uniform index 22...

diagram 4-faces 202 Cells 1560 Faces 4240 Edges 4800 Vertices 1920 Vertex figure Irregular 5-cell Coxeter group B5 [4,3,3,3] Properties convex, isogonal...

In geometry, the heptagrammic-order heptagonal tiling is a regular star-tiling of the hyperbolic plane. It has Schläfli symbol of {7,7/2}. The vertex figure...

Faces 80 50{3} 30{4} Edges 90 Vertices 30 Vertex figure Square wedge Symmetry group A4, [3,3,3], order 120 Properties convex, isogonal Uniform index 3 4 5...

In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon...

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon...

\beta _{3}=1/2} , χ 3 = [ 0 , 1 ] {\displaystyle \chi _{3}=[0,1]} . The isogonal (equiangular) hexagons of the Haythornthwaite criterion containing the...

homothetic. 3. The perpendicular bisector construction can be reversed via isogonal conjugation. That is, given Q ( i + 1 ) {\displaystyle Q^{(i+1)}} , it...

In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron is a tessellation (gap-free filling with repeated shapes) of the...

lines AA', BB', CC' are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are csc ⁡ ( A + π 3 ) : csc ⁡ ( B + π...

(acronym) Heptapeton (hop) Vertex figure 5-simplex Circumradius 3 7 {\displaystyle {\sqrt {\tfrac {3}{7}}}} 0.654654 Properties convex, isogonal self-dual...

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons...

1021/cg401646t. Dunitz, J. D.; Waser, J. (1972). "The Planarity of the Equilateral Isogonal Pentagon". Elemente der Mathematik. 27: 25–32. Dunitz, J. D.; Waser, J...

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon...

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed...

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}. This tiling represents a hyperbolic...

15 {\displaystyle 4{.}4{.}15} , and therefore has the property of being isogonal and can be classified as a semi-regular heptadecahedron. In Schläfli notation...

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It covers the hyperbolic plane, which is a non-Euclidean surface...

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}. The dual to this tiling represents...

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon...

translation (two adjacent vertices are translated, mapping the faces), isogonal (opposite angles at every vertex is equal or complementary), orthogonal...

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}. A half symmetry coloring...

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex....

In 2-dimensional hyperbolic geometry, the infinite-order hexagonal tiling is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are ideal...

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}. A half symmetry [1+,4...

In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}...

Isogonal polytope with uniform facets...

In geometry, the snub pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}. Drawn in chiral pairs, with...

In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}. John H. Conway, Heidi...

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}. The dual to this tiling represents...

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically...

In geometry, an apeirogonal antiprism or infinite antiprism is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron...

3 {\displaystyle {\tilde {C}}_{3}} , [4,3,4] Dual Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell: Properties isogonal, isotoxal, isochoric...

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual. This tiling represents...

Vertices 15 Vertex figure {}×{3,3} Coxeter group A5, [34], order 720 Dual Base point (0,0,0,0,1,1) Circumradius 0.645497 Properties convex, isogonal isotoxal...

In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,5}, constructed from...

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}. In (*∞44) symmetry...

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction...

In geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}. John...

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located...

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232...

In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular...

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}. There are two uniform...

In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}. Drawn in chiral pairs, with...

In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of...

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex...

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}. The dual tiling represents...

In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{∞,∞}. It has 3 equilateral triangles...

regular face, which alternate around each vertex. Their vertex figures are isogonal polygons. Regular and quasiregular figures Right triangle domains (p p...

In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}. John H. Conway, Heidi Burgiel...

In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from...

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}. The dual of this...

In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}. The dual tiling, the...

In geometry, the snub pentapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular...

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane. There are four uniform constructions of this tiling, three of them as...

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}. A secondary construction...