Polyscheme

The 24-point 24-cell, a 4-dimensional regular polytope, contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point octahedron is a coordinate system basis frame of three perpendicular (x,y,z) axes. One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which in a unit-radius, unit-edge 24-cell is a √2 edge of the red, green, or blue 16-cell.

Polyscheme is the name given to geometric objects of any number of dimensions (polytopes) by Ludwig Schläfli, the Swiss mathematician who discovered all the regular polytopes which exist in higher dimensions of Euclidean space before 1853, at "a time when Cayley, Grassmann^[b] and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."^[1] The Wikiversity was hosted on very slow servers in those days, and other researchers also discovered the 4-polytopes before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project.^[c] H.S.M. Coxeter is its founding editor, whose 1948 book Regular Polytopes tells the whole story of the project.

Polyscheme learning project

The Polyscheme project is intended to be a series of wiki-format articles on the regular polytopes, the fourth spatial dimension, and the general dimensional analogy of Euclidean and spherical spaces of any number of dimensions. This series of articles expands the corresponding Wikipedia encyclopedia articles, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop up in context.^[d]

Some of what is in these companion articles is the result of original research that may not have been peer reviewed yet, and so has the status of opinion as of this date of publication, and some of it is commentary, not essential fact. The commentary and recent research is precisely the difference between an expanded Wikiversity learning project article and the corresponding Wikipedia encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it.

Most project articles are an annotated and expanded version of the Wikipedia article, which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article. Their banner indicates that they are only a commentary on it, and participants are directed to the Wikipedia article by a "see also" link below the banner.

Active research

Polyschemes have been a subject of active and ongoing research since their discovery before 1853 by the Swiss researcher Ludwig Schläfli, who was the first person to extend Euclidean geometry by analogy beyond three dimensions. But for the first 50 years of the subject's history Schläfli's paper was unpublished, entirely inaccessible to other researchers. Even after its posthumous publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicians who could read that specialized language. During this period other researchers also discovered higher-dimensional Euclidean space. H.S.M. Coxeter finally made the subject widely accessible in his 1948 book Regular Polytopes, which synthesized all the research that had been done since Schläfli and added Coxeter's discoveries, including his invention of the theory of reflecting symmetry groups, the group theory mathematics that underlies geometry. Since then, Coxeter's book has been the definitive work on Euclidean geometry in n-dimensional space, and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new observations to it.

Polyscheme project articles

The following Wikiversity articles comprise the Polyscheme collection of Wikipedia companion articles at present. Please add your research contributions to them, and consider adding to this list of Polyscheme companion articles, if you have a research contribution to make to another relevant Wikipedia article, that would not be appropriate in the encyclopedia.

Regular convex 4-polytopes

Uniform 3-polytopes

Kinematics of the cuboctahedron

Other conceptual objects

Schläfli double six

Sequence of regular 4-polytopes

Sequence of 6 regular convex 4-polytopes of radius √2
Symmetry group	A₄	B₄		F₄	H₄
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell Hyper-cuboctahedron 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2	𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices^[e]	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 {3, 3}	16 {3, 3}	8 {4, 3}	24 {3, 4}	600 {3, 3}	120 {5, 3}
Tori	5 {3, 3}	8 {3, 3} x 2	4 {4, 3} x 2	6 {3, 4} x 4	30 {3, 3} x 20	10 {5, 3} x 12
Inscribed	120 in 120-cell	675 in 120-cell	2 16-cells	3 8-cells	25 24-cells	10 600-cells
Great polygons		2 squares x 3^[f]	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons^[g]	1 pentagon x 2	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Edge length^[h]	${\sqrt {5}}\approx 2.236$	${\sqrt {4}}$	${\sqrt {2}}\approx 1.414$	${\sqrt {2}}\approx 1.414$	${\sqrt {\tfrac {2}{\phi ^{2}}}}\approx 0.874$	${\sqrt {\tfrac {1}{\phi ^{4}}}}\approx 0.382$
Isocline chord^[i]	${\sqrt {3}}\approx 1.732$	${\sqrt {4}}$	${\sqrt {6}}\approx 2.449$	${\sqrt {6}}\approx 2.449$	${\sqrt {2}}\approx 1.414$	${\sqrt {\tfrac {1}{\phi ^{2}}}}\approx 0.618$
Isoclinic ratio^[j]	${\sqrt {3}}/{\sqrt {5}}\approx 0.775$	${\sqrt {1}}$	${\sqrt {3}}\approx 1.732$	${\sqrt {3}}\approx 1.732$	${\sqrt {\phi +1}}\approx 1.618$	${\sqrt {\phi +1}}\approx 1.618$
Long radius	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$	${\sqrt {2}}$
Edge radius	${\sqrt {\tfrac {3}{4}}}\approx 0.866$	${\sqrt {1}}$	${\sqrt {3}}\approx 1.732$	${\sqrt {\tfrac {3}{2}}}\approx 1.225$	${\sqrt {\tfrac {\phi {\sqrt {5}}}{4}}}\approx 1.345$	${\sqrt {\tfrac {3\phi ^{2}}{4}}}\approx 1.401$
Face radius	${\sqrt {\tfrac {25}{43}}}\approx 0.762$	${\sqrt {\tfrac {2}{3}}}\approx 0.816$	${\sqrt {1}}$	${\sqrt {\tfrac {4}{3}}}\approx 1.155$	${\sqrt {\tfrac {5\phi ^{2}}{23}}}\approx 0.754$	${\sqrt {\tfrac {\phi ^{3}}{4{\sqrt {5/2}}}}}\approx 0.818$
Short radius	${\sqrt {\tfrac {1}{8}}}\approx 0.354$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {1}}$	${\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309$	${\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309$
Area	$10\left({\tfrac {5{\sqrt {3}}}{4}}\right)\approx 21.651$	$32\left({\sqrt {3}}\right)\approx 55.425$	$48$	$96\left({\sqrt {\tfrac {3}{4}}}\right)\approx 83.138$	$1200\left({\tfrac {2{\sqrt {3}}}{4\phi ^{2}}}\right)\approx 396.95$	$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{4\phi ^{4}}}\right)\approx 180.73$
Volume	$5\left({\tfrac {5{\sqrt {10}}}{12}}\right)\approx 6.588$	$16\left({\tfrac {2{\sqrt {2}}}{3}}\right)\approx 15.085$	$8{\sqrt {8}}\approx 22.627$	$24\left({\tfrac {4}{3}}\right)=32$	$600\left({\tfrac {4}{12\phi ^{3}}}\right)\approx 47.214$	$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}}}\right)\approx 51.246$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\sqrt {5}}\right)^{4}\approx 2.329$	${\tfrac {8}{3}}\approx 2.666$	$4$	$8$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 15.451$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 16.770$

Sequence of 6 regular convex 4-polytopes of radius 1
Symmetry group	A₄	B₄		F₄	H₄
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell Hyper-cuboctahedron 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2	𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices^[e]	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 {3, 3}	16 {3, 3}	8 {4, 3}	24 {3, 4}	600 {3, 3}	120 {5, 3}
Tori	5 {3, 3}	8 {3, 3} x 2	4 {4, 3} x 2	6 {3, 4} x 4	30 {3, 3} x 20	10 {5, 3} x 12
Inscribed	120 in 120-cell	675 in 120-cell	2 16-cells	3 8-cells	25 24-cells	10 600-cells
Great polygons		2 squares x 3^[f]	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons^[g]	1 pentagon x 2	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Edge length^[h]	${\sqrt {\tfrac {5}{2}}}\approx 1.581$	${\sqrt {2}}\approx 1.414$	${\sqrt {1}}$	${\sqrt {1}}$	${\sqrt {2-\phi }}\approx 0.618$	${\sqrt {\tfrac {1}{2\phi ^{4}}}}\approx 0.270$
Isocline chord^[i]	${\sqrt {\tfrac {3}{2}}}\approx 1.225$	${\sqrt {2}}\approx 1.414$	${\sqrt {3}}\approx 1.732$	${\sqrt {3}}\approx 1.732$	${\sqrt {1}}$	${\sqrt {\tfrac {1}{2\phi ^{2}}}}\approx 0.437$
Isoclinic ratio^[j]	${\sqrt {3}}/{\sqrt {5}}\approx 0.775$	${\sqrt {1}}$	${\sqrt {3}}\approx 1.732$	${\sqrt {3}}\approx 1.732$	${\sqrt {\phi +1}}\approx 1.618$	${\sqrt {\phi +1}}\approx 1.618$
Long radius	${\sqrt {1}}$	${\sqrt {1}}$	${\sqrt {1}}$	${\sqrt {1}}$	${\sqrt {1}}$	${\sqrt {1}}$
Edge radius	${\sqrt {\tfrac {3}{8}}}\approx 0.612$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {3}{4}}}\approx 0.866$	${\sqrt {\tfrac {3}{4}}}\approx 0.866$	${\sqrt {\tfrac {\phi {\sqrt {5}}}{4}}}\approx 0.951$	${\sqrt {\tfrac {3\phi ^{2}}{8}}}\approx 0.991$
Face radius	${\sqrt {\tfrac {25}{86}}}\approx 0.539$	${\sqrt {\tfrac {1}{3}}}\approx 0.577$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {2}{3}}}\approx 0.816$	${\sqrt {\tfrac {5\phi ^{2}}{46}}}\approx 0.533$	${\sqrt {\tfrac {\phi ^{3}}{8{\sqrt {5/2}}}}}\approx 0.579$
Short radius	${\sqrt {\tfrac {1}{16}}}=0.25$	${\sqrt {\tfrac {1}{4}}}=0.5$	${\sqrt {\tfrac {1}{4}}}=0.5$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area	$10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$	$32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$	$24$	$96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$	$1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$	$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume	$5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$	$16\left({\tfrac {1}{3}}\right)\approx 5.333$	$8$	$24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$	$600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$	$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$	${\tfrac {2}{3}}\approx 0.667$	$1$	$2$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$

Notes

↑ Grassmann moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.
↑ In 1844, Hermann Grassmann proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once geometry is put into the algebraic form he advocated now known as the Grassmannian, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.^[a]
↑ Reinhold Hoppe's German word polytop was introduced into English by Alicia Boole Stott, who like Hoppe and Thorold Gosset rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original polyschem, with its echo of schema as in the configurations of information structures, seems even more fitting in its generality than polytope -- perhaps analogously as information software (programming) is even more general than information hardware (computers).
↑
The inventor Arthur Fry with a Post-it note on his forehead.
If you hover the cursor over a footnote it will pop up in a floating box like a post-it note, so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the footnote reference, you can read a larger note like this one where it appears in the Notes section of the article, like a mini-article within the article. It may occur in other Polyscheme project articles as well. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a twisty little maze of passages! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.
↑ ^5.0 ^5.1 The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.^[2] Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4-content within the same radius.^[3] The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in the ascending sequence that begins with the 5-point (5-cell) 4-polytope and ends with the 600-point (120-cell) 4-polytope.
↑ ^6.0 ^6.1 In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
↑ ^7.0 ^7.1 Coxeter describes the helical Petrie polygons of regular 4-polytopes. He begins by noting that the regular tesselations of 3-space (which may be viewed as "flat" 4-polytopes) have the same kind of helical Petrie polygons as spherical 4-polytopes:
Among the vertices and edges of a regular honeycomb $\{p,q,r\}$ we can pick out a new kind of Petrie polygon in which every three consecutive edges belong to the Petrie polygon of a cell but no four consecutive edges belong to the same cell. ... The isometry which takes us one step along the Petrie polygon, being conjugate to the product of half-turns about two opposite edges of the characteristic tetrahedron, is the product of half-turns about two skew lines, that is, a twist: the product of a translation along a line $l$ (which measures the shortest distance between two skew lines) and a rotation about the same line. Thus the Petrie polygon is a "helical" polygon: its edges are the chords of a helix. This description is valid in hyperbolic space as well as in Euclidean space.
In spherical space, $l$ is, of course, a great circle, the "translation" along it is a rotation about a polar great circle, and the twist is a compound rotation [double rotation]: the product of two rotations whose axes are polar great circles (lying in completely orthogonal planes of the Euclidean 4-space). Let $h$ denote the period of this compound rotation, so that the Petrie polygon is a skew $h$ -gon.^[4]
↑ ^8.0 ^8.1 A procedure to construct each of these 4-polytopes from the 4-polytope to its left (its predecessor) preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The successor edge length will always be less unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.
↑ ^9.0 ^9.1 The 4-space distance by which each vertex is displaced in each step of the characteristic isoclinic (equi-angled) double rotation. Each vertex is displaced within its moving invariant central plane by a 2-space distance of one edge length. The invariant rotation planes are a Clifford parallel subset of all the edge central planes, but they include all the vertices. Every edge is displaced to another edge that lies the characteristic chordal distance away, whether or not the edge lies in an invariant plane during the rotation.
↑ ^10.0 ^10.1 The ratio of the isocline chord to the edge length is a characteristic constant independent of the metric unit (long radius).

Citations

↑ Coxeter 1973, pp. 141-144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."
↑ Coxeter 1973, p. 136, §7.8 The enumeration of possible regular figures.
↑ Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
↑ Coxeter 1970, p. 25, Twisted Honeycombs, §11. The Petrie polygon of a honeycomb.

References

Kepler, Johannes (1619). Harmonices Mundi (The Harmony of the World). Johann Planck.
Gosset, Thorold (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions". Messenger of Mathematics (Macmillan).
Boole Stott, Alicia (1910), Geometrical deduction of semiregular from regular polytopes and space fillings (PDF), Amsterdam: Johannes Muller, pp. 12–45
Schoute, Pieter Hendrik (1911), Analytical treatment of the polytopes regularly derived from the regular polytopes, Amsterdam: Johannes Muller, pp. 114–197
Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
Coxeter, H.S.M. (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, Providence, Rhode Island: American Mathematical Society, 4
Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
Coxeter, H.S.M. (1995). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C. et al.. eds. Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.). Wiley-Interscience Publication. ISBN 978-0-471-01003-6. https://www.wiley.com/en-us/Kaleidoscopes%3A+Selected+Writings+of+H+S+M+Coxeter-p-9780471010036.
- (Paper 3) H.S.M. Coxeter, Two aspects of the regular 24-cell in four dimensions
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. 6. University of Toronto Studies (Mathematical Series).
Coxeter, H.S.M. (1968). The Beauty of Geometry: Twelve Essays (2nd ed.). New York: Dover.
Coxeter, H.S.M. (1989). "Trisecting an Orthoscheme". Computers Math. Applic. 17 (1–3): 59–71. doi:10.1016/0898-1221(89)90148-X.
Coxeter, H.S.M.; Shephard, G.C. (1992). "Portraits of a family of complex polytopes". Leonardo 25 (3/4): 239–244. doi:10.2307/1575843.

Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie. Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8. https://archive.org/details/shapingspaceexpl00sene.
Banchoff, Thomas (1989). "Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin. Computers in geometry and topology. New York and Basel: Marcel Dekker. pp. 57–62.
Borovik, Alexandre (2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich. The Coxeter Legacy. Providence, Rhode Island: American Mathematical Society. pp. 17–43. ISBN 978-0821837221. https://www.academia.edu/26091464.
Chilton, B. L. (September 1964). "On the projection of the regular polytope {5,3,3} into a regular triacontagon". Canadian Mathematical Bulletin 7 (3): 385–398. doi:10.4153/CMB-1964-037-9.
Christie, David Brooks (2022), "Kinematics of the cuboctahedron", Polyscheme, Wikiversity
Christie, David Brooks, ed. (2024), "The Polyscheme Project", Wikiversity
Conway, John; Guy, Michael (1965). "Four-Dimensional Archimedean Polytopes". Proceedings of the Colloquium on Convexity at Copenhagen.
Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. ISBN 978-1-56881-220-5.
Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv:1903.06971 [math.MG].
Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras (Springer Science and Business Media) 31 (3). doi:10.1007/s00006-021-01139-2.
Dechant, Pierre-Philippe (2017). "The E8 Geometry from a Clifford Perspective". Advances in Applied Clifford Algebras 27: 397–421. doi:10.1007/s00006-016-0675-9.
Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry 20 (3): 433–444. doi:10.1515/advgeom-2020-0005.
Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras 29 (44). doi:10.1007/s00006-019-0960-5.
Egan, Greg (23 December 2021). "Symmetries and the 24-cell". gregegan.net. Retrieved 10 October 2022.
Goucher, A.P. (19 November 2019), "Spin groups", Complex Projective 4-Space
Goucher, A.P. (1 October 2020), "Subsumptions of regular polytopes", Complex Projective 4-Space
Johnson, Norman (2018), Geometries and Transformations, Cambridge: Cambridge University Press, ISBN 978-1-107-10340-5
Johnson, Norman (1991), Uniform Polytopes (Manuscript ed.)
Johnson, Norman (1966), The Theory of Uniform Polytopes and Honeycombs (Ph.D. ed.)
Kappraff, Jay; Adamson, Gary (2000). "A Fresh Look at Number". BRIDGES: Mathematical Connections in Art, Music, and Science. https://archive.bridgesmathart.org/2000/bridges2000-255.html.
Kappraff, Jay; Adamson, Gary (2003). "The Relationship of the Cotangent function to Special Relativity Theory, Silver Means, p-cycles, and Chaos Theory". Forma. https://www.researchgate.net/publication/228566360_The_Relationship_of_the_Cotangent_Function_to_Special_Relativity_Theory_Silver_Means_p-cycles_and_Chaos_Theory.
Kappraff, Jay; Jablan, Slavik; Adamson, Gary; Sazdanovich, Radmila (2004). "Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices". Forma 19: 367-387. https://archive.bridgesmathart.org/2005/bridges2005-369.pdf.
Kappraff, Jay; Adamson, Gary (2004). "Polygons and Chaos". Dynamical Systems and Geometric Theories. https://archive.bridgesmathart.org/2001/bridges2001-67.pdf.
Kim, Heuna; Rote, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Klitzing, Richard (2025). "Polytopes and their Incidence Matrices".
Koca, Mehmet; Al-Ajmi, Mudhahir; Koc, Ramazan (November 2007). "Polyhedra obtained from Coxeter groups and quaternions". Journal of Mathematical Physics 48 (11): 113514. doi:10.1063/1.2809467. https://www.researchgate.net/publication/234907424.
Koca, Mehmet; Ozdes Koca, Nazife; Al-Ajmi, Mudhahir (02 2011). 4D Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions. https://arxiv.org/pdf/1102.1132.
Koca, Mehmet; Ozdes Koca, Nazife; Al-Ajmi, Mudhahir (06 2011). Branching of the W(H2) Polytopes and Their Dual Polytopes under the Coxeter Groups W(A4) and W(H3) Represented by Quaternions. https://arxiv.org/pdf/1106.2957.
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications 434 (4): 977–989. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. https://arxiv.org/pdf/0906.2109.
Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)". Int. J. Geom. Methods Mod. Phys. 09 (8). doi:10.1142/S0219887812500685. http://arxiv-web3.library.cornell.edu/abs/1106.3433.
Koca, Nazife; Al-Mukhaini, Aida; Al Qanobi, Amal (06 2016). 4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices. https://arxiv.org/pdf/1606.02445.
Koca, Mehmet; Al-Mukhaini, Aida (2016-12-01). "Symmetry of the Pyritohedron and Lattices". Sultan Qaboos University Journal for Science [SQUJS] 21 (2): 139. doi:10.24200/squjs.vol21iss2pp139-149. https://squjs.squ.edu.om/cgi/viewcontent.cgi?article=1214&context=squjs.
Lemmens, P.W.H.; Seidel, J.J. (1973). "Equi-isoclinic subspaces of Euclidean spaces". Indagationes Mathematicae (Proceedings) 76 (2): 98–107. doi:10.1016/1385-7258(73)90042-5. ISSN 1385-7258. https://dx.doi.org/10.1016/1385-7258%2873%2990042-5.
Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry 2 (3): 1423–1449. doi:10.3390/sym2031423.
Mebius, Johan, ed. (2015), "SO(4): Rotations in 4-dimensional Euclidean space", Wikipedia
Mebius, Johan (July 2015) [11 Jan 1994]. Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics (Thesis). Delft University of Technology. doi:10.13140/RG.2.1.3310.3205.
Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes". International Journal of Space Structures 5 (3–4): 309–323. doi:10.1177/026635119000500312.
Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications". Adv. Appl. Clifford Algebras 27: 523–538. doi:10.1007/s00006-016-0683-9. https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf.
Polo-Blanco, Irene (2007). "§5 Alicia Boole Stott and four-dimensional polytopes". Theory and history of geometric models. University of Groningen. https://pure.rug.nl/ws/portalfiles/portal/2803503/c5.pdf.
Ruen, Tom, ed. (2011), "Triacontagon", Wikipedia
Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E 5: 575–582. doi:10.1007/s101890170040. https://www.researchgate.net/publication/260046074.
Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A 42 (46): 465209. doi:10.1088/1751-8113/42/46/465209. https://stacks.iop.org/JPhysA/42/465209.
Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494.
Steinbach, Peter (2000). "Sections Beyond Golden". Bridges: Mathematical Connections in Art, Music and Science (2000): 35-44. https://archive.bridgesmathart.org/2000/bridges2000-35.pdf.
Stillwell, John (January 2001). "The Story of the 120-Cell". Notices of the AMS 48 (1): 17–25. https://www.ams.org/notices/200101/fea-stillwell.pdf.
Sullivan, John M. (1991). "Generating and Rendering Four-Dimensional Polytopes". Mathematica Journal 1 (3): 76–85. http://isama.org/jms/Papers/dodecaplex/.
Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8. https://archive.org/details/generalizedcliff0000tyrr.
van Ittersum, Clara (2020). Symmetry groups of regular polytopes in three and four dimensions. http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89.
Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A 43 (10): 105304. doi:10.1088/1751-8113/43/10/105304.
Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell". Foundations of Physics 44 (10): 1085–1095. doi:10.1007/s10701-014-9830-0.
Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering 8 (3): 836–854. doi:10.1093/jcde/qwab018.

[1] Grassmann moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.

[Grassmann-2] In 1844, Hermann Grassmann proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once geometry is put into the algebraic form he advocated now known as the Grassmannian, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.^[a]

[4] Reinhold Hoppe's German word polytop was introduced into English by Alicia Boole Stott, who like Hoppe and Thorold Gosset rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original polyschem, with its echo of schema as in the configurations of information structures, seems even more fitting in its generality than polytope -- perhaps analogously as information software (programming) is even more general than information hardware (computers).

[explanatory_notes-5] 
The inventor Arthur Fry with a Post-it note on his forehead.
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[4-polytopes_ordered_by_size_and_complexity-6] 5.0 ^5.1 The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.^[2] Each greater polytope in the sequence is rounder than its predecessor, enclosing more 4-content within the same radius.^[3] The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in the ascending sequence that begins with the 5-point (5-cell) 4-polytope and ends with the 600-point (120-cell) 4-polytope.

[Six_orthogonal_planes_of_the_Cartesian_basis-7] 6.0 ^6.1 In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.

[Petrie_polygon_of_a_honeycomb-8] 7.0 ^7.1 Coxeter describes the helical Petrie polygons of regular 4-polytopes. He begins by noting that the regular tesselations of 3-space (which may be viewed as "flat" 4-polytopes) have the same kind of helical Petrie polygons as spherical 4-polytopes:
Among the vertices and edges of a regular honeycomb $\{p,q,r\}$ we can pick out a new kind of Petrie polygon in which every three consecutive edges belong to the Petrie polygon of a cell but no four consecutive edges belong to the same cell. ... The isometry which takes us one step along the Petrie polygon, being conjugate to the product of half-turns about two opposite edges of the characteristic tetrahedron, is the product of half-turns about two skew lines, that is, a twist: the product of a translation along a line $l$ (which measures the shortest distance between two skew lines) and a rotation about the same line. Thus the Petrie polygon is a "helical" polygon: its edges are the chords of a helix. This description is valid in hyperbolic space as well as in Euclidean space.
In spherical space, $l$ is, of course, a great circle, the "translation" along it is a rotation about a polar great circle, and the twist is a compound rotation [double rotation]: the product of two rotations whose axes are polar great circles (lying in completely orthogonal planes of the Euclidean 4-space). Let $h$ denote the period of this compound rotation, so that the Petrie polygon is a skew $h$ -gon.^[4]

[edge_length_of_successor-9] 8.0 ^8.1 A procedure to construct each of these 4-polytopes from the 4-polytope to its left (its predecessor) preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The successor edge length will always be less unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.

[isocline_chord-10] 9.0 ^9.1 The 4-space distance by which each vertex is displaced in each step of the characteristic isoclinic (equi-angled) double rotation. Each vertex is displaced within its moving invariant central plane by a 2-space distance of one edge length. The invariant rotation planes are a Clifford parallel subset of all the edge central planes, but they include all the vertices. Every edge is displaced to another edge that lies the characteristic chordal distance away, whether or not the edge lies in an invariant plane during the rotation.

[isocline_chord_to_edge_length_ratio-11] 10.0 ^10.1 The ratio of the isocline chord to the edge length is a characteristic constant independent of the metric unit (long radius).

[FOOTNOTECoxeter1973141-144§7._Ordinary_Polytopes_in_Higher_Space;_§7.x._Historical_remarks-3] Coxeter 1973, pp. 141-144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."

[FOOTNOTECoxeter1973136§7.8_The_enumeration_of_possible_regular_figures-12] Coxeter 1973, p. 136, §7.8 The enumeration of possible regular figures.

[FOOTNOTECoxeter1973292-293Table_I(ii):_The_sixteen_regular_polytopes_{''p,q,r''}_in_four_dimensions-13] Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.

[FOOTNOTECoxeter197025''Twisted_Honeycombs'',_§11._The_Petrie_polygon_of_a_honeycomb-14] Coxeter 1970, p. 25, Twisted Honeycombs, §11. The Petrie polygon of a honeycomb.

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