^ P 0と P 1 は同じ超平面（同じ中心立方八面体）にあるので、もう一方の分離角は 0 です。[aw]
^ V 0と V 2は、この回転等傾斜線の測地線経路上で2 √ 3弦離れていますが、これはそれらの間の最短測地線経路ではありません。24 セルでは、2 つの頂点が1 √ 3弦より離れることは、それらが√ 4離れている反対側の頂点でない限り、不可能です。[ah] V 0と V 2 は、他の等傾斜線上では1 √ 3弦離れており、ある大きな六角形ではわずか√ 1離れています。V 0と V 2の間では、等傾斜回転が 24 セルの周りを 2 √ 3弦にわたって長い道のりを進み、わずか√ 1離れた頂点に到達しています。より一般的には、等傾斜線が測地線であるのは、連続する頂点間の距離が、それらを接続するある回転におけるそれらの 2 つの頂点間の最短距離であるためですが、3 次元球面上では、より短い別の回転がある場合があります。測地線に沿った 2 つの頂点間の経路は、必ずしもそれらの間の最短距離であるとは限りません (通常の大円測地線上でも同様です)。
^ P 0と P 2は、どちらの分離角度でも 60° 離れています。 [aw]クリフォード平行面は等傾斜面 (つまり、2 つの等しい角度で分離されている) であり、対応する頂点はすべて同じ距離離れています。V 0と V 2は2 つの√ 3弦離れていますが、[cr] P 0と P 2 は1 つの√ 1辺だけ離れています(最も近い頂点のすべてのペアで)。
^ abc 等傾斜回転の元の大六角形平面 P 0 の任意の頂点 V 0から出発して、最初の頂点 V 1に到達するまでには、別の六角形平面 P 1にある√ 3弦に沿って 120 度かかります。P 1 はP 0に対して60 度の角度で傾斜しています。 [cq] 2 番目の頂点 V 2に到達するまでには、 P 0とクリフォード平行な別の六角形平面 P 2にある2 番目の√ 3弦に沿って V 1から 120 度かかります。[cs] (V 1 は交差する平面 P 1と P 2 の両方に位置し、V 0 はP 0と P 1 の両方に位置していることに注意してください。しかし、P 0と P 2には共通の頂点がなく、交差しません。) 3 番目の頂点 V 3は、P 1とクリフォード平行な別の六角形平面 P 3に位置する3 番目の√ 3弦に沿って V 2から 120度先にあります。V 0と V 3 は隣接する頂点で、√ 1離れています。[ct] 3 つの√ 3弦は、異なる 8 セルにあります。[t] V 0から V 3 は360° 等傾斜回転であり、24 セルの二重ループ六角形2クリフォード多角形の半分です。[cl]
^Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.[cl]
^ a bIn this orthogonal projection of the 24-point 24-cell to a {12/4}=4{3} dodecagram, each point represents two vertices, and each line represents multiple √3 chords. Each disjoint triangle can be seen as a skew {6/2} hexagram with √3 edges: two open skew triangles with their opposite ends connected in a Möbius loop with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew hexagram isoclines are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) isoclinic rotation.[at] The 4 Clifford parallel great hexagons of the fibration[r] are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels and 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.[eo] Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel great triangles, where two opposing great triangles lie in the same great hexagon central plane, so a fibration of 4 Clifford parallel great hexagon planes is represented.[r] This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the same fibration as 4 great hexagons.
^The isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° = √3 away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^ a bIn this orthogonal projection of the 24-point 24-cell to a {12/4}=4{3} dodecagram, each point represents two vertices, and each line represents multiple √3 chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel great triangles, where two opposing great triangles lie in the same great hexagon central plane, so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).[r]
^Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.[cl]
^ a b cIn this orthogonal projection of the 24-point 24-cell to a {12/2}=2{6} dodecagram, each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} dodecagon, a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a Möbius loop with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) isoclinic rotation.[at] The 4 Clifford parallel great hexagons of the fibration[r] are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels and 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.[es] Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.[r] This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the same fibration as 4 great hexagons.
^At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.
^The isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° = √1 away), without passing through any intervening vertices.[eu] Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° = √0 away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° = √4 away),[ek] without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The edges and 4𝝅 characteristic rotations of the 16-cell lie in the great square central planes. Rotations of this type are an expression of the symmetry group. The edges and 4𝝅 characteristic rotations of the 24-cell lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the symmetry group.
^Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.[aw] Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.
^ a b cThis hybrid isoclinic rotation carries the two kinds of central planes to each other: great square planes characteristic of the 16-cell and great hexagon (great triangle) planes characteristic of the 24-cell.[ey] This is possible because some great hexagon planes lie Clifford parallel to some great square planes.[ez]
^The isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° = √1 away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.[fa] Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.[x] Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).
^Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups and (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.
^The isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° = √3 away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.[fa] Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° = √0 away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^A quaternion Cartesian coordinate designates a vertex joined to a top vertex by one instance of a distinct chord. The conventional top vertex of a unit radius 4-polytope in cell-first orientation is . Thus e.g. designates a √2 chord of 90° arc-length. Each such distinct chord is an edge of a distinct great circle polygon, in this example a great square, intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete Hopf fibration that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate is thus representative of the 6 disjoint great squares pictured, a quaternion group[ek] which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.[j]
^The representative coordinate is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's "reflecting circles") run through cell centers rather than through vertices, and quaternion group corresponds to a set of those. However, also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).[fg]
^The isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° = √4 away,[ek] without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, which in this rotation is the completely orthogonal plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^Icositetragon {24/9}=3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.This orthogonal projection of a 24-cell to a 24-gram {24/9}=3{8/3} exhibits 3 disjoint octagram {8/3} isoclines of a 16-cell, each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.
^In this orthogonal projection of the 24-point 24-cell to a {12/3}=3{4} dodecagram, each point represents two vertices, and each line represents multiple √2 chords. Each disjoint square can be seen as a skew {8/3} octagram with √2 edges: two open skew squares with their opposite ends connected in a Möbius loop with a circumference of 4𝝅, visible in the {24/9}=3{8/3} orthogonal projection.[fj] The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint skew octagram isoclines are the circular vertex paths characteristic of an isoclinic rotation in great square planes, in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels and 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.[j] This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the same fibration as 6 squares.
^At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.
^The isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° = √2 away, without passing through any intervening vertices.[fl] Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° = √0 away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° = √1 away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.[fa] Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° = √4 away,[ek] without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, which in this rotation is the completely orthogonal plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.
^The isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° = √2 away, without passing through any intervening vertices.[fq] Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^The rotation is the identity operation of the 24-cell, in which no points move.
^The rotation is the central inversion of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° = √4 away,[ek] without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, which in this rotation is the completely orthogonal plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.
^A right rotation is performed by rotating the left and right planes in the "same" direction, and a left rotation is performed by rotating left and right planes in "opposite" directions, according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes. Left and right rotations are chiral enantiomorphous shapes (like a pair of shoes), not opposite rotational directions. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.[cf]
Citations
^Coxeter 1973, p. 118, Chapter VII: Ordinary Polytopes in Higher Space.
^Johnson 2018, p. 249, 11.5.
^Ghyka 1977, p. 68.
^Coxeter 1973, p. 289, Epilogue; "Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below."
^Coxeter 1995, p. 25, (Paper 3) Two aspects of the regular 24-cell in four dimensions.
^Coxeter 1968, p. 70, §4.12 The Classification of Zonohedra.
^Coxeter 1973, p. 136, §7.8 The enumeration of possible regular figures.
^ a bCoxeter 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 1973, p. 302, Table VI (ii): 𝐈𝐈 = {3,4,3}: see Result column
^Coxeter 1973, p. 156, §8.7. Cartesian Coordinates.
^Coxeter 1973, pp. 145–146, §8.1 The simple truncations of the general regular polytope.
^Waegell & Aravind 2009, pp. 4–5, §3.4 The 24-cell: points, lines and Reye's configuration; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.
^Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column a.
^Stillwell 2001, p. 17.
^Tyrrell & Semple 1971, pp. 5–6, §3. Clifford's original definition of parallelism.
^Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
^ a bStillwell 2001, p. 24.
^Copher 2019, p. 6, §3.2 Theorem 3.4.
^Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)".
^Coxeter 1973, p. 153, 8.5. Gosset's construction for {3,3,5}: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."
^Coxeter 1973, p. 304, Table VI(iv) II={5,3,3}: Faceting {5,3,3}[120𝛼4]{3,3,5} of the 120-cell reveals 120 regular 5-cells.
^Egan 2021, animation of a rotating 24-cell: red half-integer vertices (tesseract), yellow and black integer vertices (16-cell).
^ a b cCoxeter 1973, p. 150, Gosset.
^Coxeter 1973, p. 148, §8.2. Cesaro's construction for {3, 4, 3}..
^Coxeter 1973, p. 302, Table VI(ii) II={3,4,3}, Result column.
^Coxeter 1973, pp. 149–150, §8.22. see illustrations Fig. 8.2A and Fig 8.2B
^Coxeter 1995, p. 29, (Paper 3) Two aspects of the regular 24-cell in four dimensions; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±1/2, ±1/2, 0, 0)]".
^Coxeter 1973, p. 147, §8.1 The simple truncations of the general regular polytope; "At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....[i] In fact, the [various] radii 0𝑹, 1𝑹, 2𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈0, 𝐈𝐈1, 𝐈𝐈2, ... of the original polytope."
^ a bKepler 1619, p. 181.
^van Ittersum 2020, pp. 73–79, §4.2.
^Coxeter 1973, p. 269, §14.32. "For instance, in the case of ...."
^van Ittersum 2020, p. 79.
^Coxeter 1973, p. 150: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the . (Their centres are the mid-points of the 24 edges of the .)"
^Coxeter 1973, p. 12, §1.8. Configurations.
^Coxeter 1973, p. 120, §7.2.: "... any n+1 points which do not lie in an (n-1)-space are the vertices of an n-dimensional simplex.... Thus the general simplex may alternatively be defined as a finite region of n-space enclosed by n+1 hyperplanes or (n-1)-spaces."
^van Ittersum 2020, p. 78, §4.2.5.
^Stillwell 2001, p. 18-21.
^Egan 2021; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.
^Stillwell 2001, p. 22.
^Koca, Al-Ajmi & Koc 2007.
^Coxeter 1973, p. 163: Coxeter notes that Thorold Gosset was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.
^ a bCoxeter 1973, p. 156: "...the chess-board has an n-dimensional analogue."
^Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹4.
^Coxeter 1973, p. 119, §7.1. Dimensional Analogy: "For instance, seeing that the circumference of a circle is 2π r, while the surface of a sphere is 4π r 2, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2π2r3."
^Kim & Rote 2016, p. 6, §5. Four-Dimensional Rotations.
^Perez-Gracia & Thomas 2017, §7. Conclusions; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change."
^Perez-Gracia & Thomas 2017.
^Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping.
^Coxeter 1995, pp. 30–32, (Paper 3) Two aspects of the regular 24-cell in four dimensions; §3. The Dodecagonal Aspect;[cg] Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the 120-cell, a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.
^Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations.
^Kim & Rote 2016, pp. 7–10, §6. Angles between two Planes in 4-Space.
^Coxeter 1973, p. 141, §7.x. Historical remarks; "Möbius realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by H. G. Wells in The Plattner Story."
^Feynman & Weinberg 1987, The reason for antiparticles.
^Mebius 2015, pp. 2–3, Motivation; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the Philippine wine dance or Binasuan and performed by physicist Richard P. Feynman during his Dirac memorial lecture 1986[53] to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is."
^Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."
^Kim & Rote 2016, pp. 8–9, Relations to Clifford parallelism.
^Kim & Rote 2016, p. 8, Left and Right Pairs of Isoclinic Planes.
^Tyrrell & Semple 1971, pp. 1–9, §1. Introduction.
^Tyrrell & Semple 1971, pp. 20–33, Clifford Parallel Spaces and Clifford Reguli.
^Coxeter 1973, pp. 292–293, Table I(i): Octahedron.
^Kim & Rote 2016, pp. 14–16, §8.3 Properties of the Hopf Fibration; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.
^Kim & Rote 2016, p. 12, §8 The Construction of Hopf Fibrations; 3.
^Tyrrell & Semple 1971, pp. 34–57, Linear Systems of Clifford Parallels.
^Coxeter 1973, pp. 292–293, Table I(ii); 24-cell h1 is {12}, h2 is {12/5}.
^Coxeter 1973, pp. 292–293, Table I(ii); 24-cell Petrie polygon h1 is {12}.
^Coxeter 1973, pp. 292–293, Table I(ii); 24-cell Petrie polygon orthogonal h2 is {12/5}, half of {24/5} as each Petrie polygon is half the 24-cell.
^Coxeter 1973, pp. 292–293, Table I(ii); "24-cell".
^Coxeter 1973, p. 139, §7.9 The characteristic simplex.
^Coxeter 1973, p. 290, Table I(ii); "dihedral angles".
^Coxeter 1973, pp. 130–133, §7.6 The symmetry group of the general regular polytope.
^Kim & Rote 2016, pp. 17–20, §10 The Coxeter Classification of Four-Dimensional Point Groups.
^Coxeter 1973, pp. 33–38, §3.1 Congruent transformations.
^Coxeter 1973, pp. 217–218, §12.2 Congruent transformations.
^Coxeter 1973, p. 138; "We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle."
^Coxeter 1970, p. 18, §8. The simplex, cube, cross-polytope and 24-cell; Coxeter studied cell rings in the general case of their geometry and group theory, identifying each cell ring as a polytope in its own right which fills a three-dimensional manifold (such as the 3-sphere) with its corresponding honeycomb. He found that cell rings follow Petrie polygons[cg] and some (but not all) cell rings and their honeycombs are twisted, occurring in left- and right-handed chiral forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.[do] Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the 24-cell honeycomb.
^Banchoff 2013, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the Clifford torus, showed how the honeycombs correspond to Hopf fibrations, and made a particular study of the 24-cell's 4 rings of 6 octahedral cells with illustrations.
Coxeter, H.S.M. (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 4, Providence, Rhode Island: American Mathematical Society
Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
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Ghyka, Matila (1977). The Geometry of Art and Life. New York: Dover Publications. ISBN 978-0-486-23542-4.
Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv:1903.06971 [math.MG].
van Ittersum, Clara (2020). Symmetry groups of regular polytopes in three and four dimensions (Thesis). Delft University of Technology.
Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A: Mathematical and Theoretical. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Egan, Greg (23 December 2021). "Symmetries and the 24-cell". gregegan.net. Retrieved 10 October 2022.
Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
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External links
24-cell animations
24-cell in stereographic projections
24-cell description and diagrams Archived 2007-07-15 at the Wayback Machine
Petrie dodecagons in the 24-cell: mathematics and animation software
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