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There are many things about the new 2022 interface that made me a bit uncomfortable on first using it, but in my experience so far the designers have made only one game changer, deal breaker change, by removing a feature I can't give up, so I will stay with the 2010 interface forever if they don't bring that feature back, at least as a user appearance preference. That's their removal of the old inline TOC at the top of the article, of course. The new pop-up sidebar TOC with its floating button is not a static TOC, it's a different feature entirely, innovative and useful in its own way (although the way its floating button always blocks the upper left corner of the page is very visually annoying, and you cannot get it out of the way no matter what you do by repositioning the page). But no matter -- that's not the deal-breaker. The pop-up sidebar TOC, whether you like it or not, isn't a TOC at the beginning of the article, which has been the signature appearance of every Wikipedia article since time immemorial.
When you open a Wikipedia article you expect to see a lede (like the abstract of a research article), followed by a table of contents showing the structure and organization of the article, giving you an instant idea of whether this article is 1 or 100 pages long, and how developed it is. As you refer to the article again and again over time, you will probably depart from that TOC to places you have discovered within the article again and again, your body developing a kind of muscle memory for the way the space inside the article branches out from the top. Your mind is learning the geometry of part of the vast space that is Wikipedia. The TOC at the top of every article illustrates one local part of that space. The TOC is the article editors' best attempt to choose a geometry for that subject that makes sense. It is editor-written content, artistry, not merely a generated index or search results; in fact it is the most important content in the article, after the lede. Sometimes it's all you read of an article (the lede and the TOC), and it tells you that you don't need to know any more. It can be collapsed or expanded, as suits your personal need of it, but surely it should not be entirely hidden in an always-collapsed pop-up sidebar.
The designers should fix this flaw in the new interface by simply bringing back the static TOC exactly as it is in the 2010 interface. The pop-up sidebar displaying the TOC can remain too, just don't display its floating button until the display is scrolled down to below the static TOC. It would also be a diplomatic policy decision (a no-brainer, really) to provide a user appearance preference for a static TOC, a pop-up TOC, or both.
The vertices of the regular icosahedron form five sets of three concentric, mutually orthogonalgolden rectangles, whose edges form Borromean rings. In a Jessen's icosahedron of unit short radius one set of these three rectangles (the set in which the Jessen's icosahedron's long edges are the rectangles' long edges) measures . These three rectangles are the shortest possible representation of the Borromean rings using only edges of the integer lattice.[1]
If we truncate eight vertices of a dodecahedron belonging to a cube we get a golden icosahedron (8 faces are equilateral triangles, the 12 others are golden triangles) which has tetrahedral symmetry (only three planes of symmetry). http://www.polyhedra-world.nc/tetra_di_.htm
Dimensions of Jessen's icosahedron. All dihedral angles are 90°. The vertices of the inscribed cube are the centers of the equilateral triangle faces. The polyhedron is a construct of the lengths √1√2√3√4√5√6 and the angles 𝝅/2𝝅/3𝝅/4.
The octahedron is unique among the Platonic solids in having an even number of faces (4 triangles) around each vertex. Consequently it is the only Platonic solid whose mirror planes run entirely through edges and do not divide any of its faces. Among the quasiregular solids, the cuboctahedron also has an even number of faces around each vertex (two triangles and two squares).
Because it is a quasiregular polyhedron, the cuboctahedron cannot be dissected into instances of a single characteristic tetrahedron that is an orthoscheme.[e] As a quasiregular polyhedron, the cuboctahedron possesses all the square and equilateral triangle faces of the cube and octahedron for which it is named. It can be dissected into the component orthoschemes of that cube and octahedron; but these are two different orthoschemes. A left-handed orthoscheme and a right-handed orthoscheme of the same kind meet at each face: two mirror-image characteristic orthoschemes of the cube meet at each square face, and two mirror-image characteristic orthoschemes of the octahedron meet at each triangular face.
A characteristic tetrahedron must represent all the symmetries that exist in the polyhedron.[f] It must be possible to construct the polyhedron by replicating its characteristic tetrahedron, so among other properties, the characteristic tetrahedron of the cuboctahedron would require edge lengths of 1/2, 1/2, √3/2, 1, √2, and √3 (as well as √4 unless the cuboctahedron could be dissected into characteristic tetrahedra with one vertex at its center). No such tetrahedron exists.
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
The same 720° isoclinic rotation takes each of its 1152 characteristic 5-cells to and through 11 other characteristic 5-cells, as an alternating sequence of left- and right-hand 4-orthoschemes (a sequence of 24 reflections), on a geodesic two-revolution orbit around the 3-sphere that covers 12 vertices (with each 5-cell occupying just one 24-cell vertex at a time).
The 12 stationary 4-orthoschemes visited by any one moving 4-orthoscheme in the course of a 720° isoclinic rotation are each cell-bonded to two others linearly, like the cars of a railroad train with alternating (4-dimensional) cars of two mirror-image shapes: the left-hand and right-hand forms of the same irregular 5-cell. The train runs on a circular geodesic track[n] which it entirely fills, so it has no first or last car. The train of 5-cells forms a Möbius ring that wraps twice around the 24-cell without intersecting itself in any point. Although it visits half the 24-cell vertices just once, it consists of only 12 of its 1152 5-cells, and comprises only one 96th of its 4-dimensional content.
Improve Our train analogy is not quite right, as train cars travel by translation not by reflection. Two reflections is a translation or rotation. A pair of adjacent 5-cells travels together by rotation. Perhaps each train car should be a left/right pair of 5-cells.
In the course of a 720° isoclinic rotation, the five vertices of each 5-cell occupy (in an alternating sequence of reflected 5-cells) the five vertex positions of 12 left-hand characteristic 5-cells and 12 right-hand characteristic 5-cells, while the orbiting 5-cell turns itself completely inside-out twice (the 5-cell itself rotating twice as it performs this orbit).[o] Two revolutions (a 720° isoclinic rotation) does not quite take the moving 4-orthoscheme back to itself, however. It requires 96 such 720° isoclinic rotations (an orbit of 192 revolutions) to visit all 1152 4-orthoschemes and return the moving 4-orthoscheme to its original orientation.[p]
Unresolved If the 12-vertex rail running through the 24-cell vertices is a closed loop on a geodesic isocline as we have claimed, how many 5-cell-disjoint sequences of 12 5-cells run along that rail? It had better be 48, or the forgoing theory, at least, is falsified. But it is not inconceivable that 48 5-cells could surround each segment of the geodesic isocline. There are 48 5-cells meeting in each octahedral facet of the 24-cell, with their 3-orthoscheme bases (48 characteristic tetrahedra of the octahedron) packed around each octahedral cell center. How many 5-cells meet at each 24-cell vertex (also octahedron) vertex)? Perhaps also 48 (by duality)? Then there would be 48 directions for a train of reflecting 5-cells to depart from, or arrive at, each vertex.
Isomorphism Notice that each 720° ring of cell-bonded 4-orthoschemes has an exterior spine (the largest-radius rail of the 5-helix circular rail track) which is a sequence of the same 24 vertices of the 24-cell. The 48 ring sequences are 48 different orderings of those 24 vertices. Each isocline geodesic is like an entire fibration of the 24-cell, all by itself, a linearization of the 4-polytope into a single wrapped-around-twice fiber that visits all 24 vertices. It is not a Hopf fibration, which ic composed of multiple parallel great circle fibers. But as in any set of fibrations, each fibration runs in a different "direction" (related to the path of its particular isoclinic rotation) by virtue of having a different ordering of the same set of vertices. In the case of an "isoclinic fibration" there are no parallel fibers. What is the relationship between Hopf fibrations and "isoclinic fibrations"? We know that isoclines cross great circles, knitting them together, and we know that the relationship (in the 24-cell) is 4:1 (4 great circles of 6 hexagons versus a Mobius loop of 2 decagons). All fibrations are the same 24 vertices and differ only in their orderings and separation into non-intersecting parallel loops, but how exactly are these characterizing properties related in the Hopf fibration versus "isoclinic fibration" cases? Is each "isoclinic fibration" geometrically isomorphic to a Hopf fibration somehow, or is an "isoclinic fibration" a thing in its own right? Of course it is the former of these two possibilities which is correct, because the Clifford parallel Hopf fibers are related by an isoclinic rotation which brings them together.[q] The Hopf fibers and the isoclines crossing between them are the warp and woof of the same fibration: that is their isomorphism. How many isoclines does each particular isoclinic rotation have in the 24-cell? Are they disjoint?
Resolved Why are we studying only isoclinic rotations re: orthoschemes? Because only in a double rotation do all the points except the 24-cell center move. Isoclinic rotation is the most symmetrical case; we study it first. But more generally, every translation or rotation in 4-space can be viewed as an isoclinic rotation by appropriate choice of reference frame; so every displacement in 4-space is an isoclinic rotation.[t]
Hi Beland. Thank you for reviewing the 600-cell article, and for your suggestion that the copious footnotes be simplified and improved, and perhaps pulled inline or moved to separate articles. The article does have a great many explanatory footnotes! The Notes section is 3/5 as large as all the rest of the article combined. That is indeed very unusual for a Wikipedia article, even a large article on a complex topic. So as the author of most of those notes, I feel it is incumbent upon me to try to explain why they are there. The heavy use of multiply-linked explanatory footnotes is certainly an outlier in the range of Wikipedia footnoting styles, but I hope to persuade you that in general they are a feature of this article, not a bug. I do agree that many of the concepts they explain deserve a separate Wikipedia article of their own, and I will be working on developing those articles in the future. It would be ideal to group the text of footnotes on the same topic into an article of its own, so the footnote references can be replaced with links.
The reason for the heavily interlinked footnotes is the special nature of the subject matter. This is an article about a 120-vertex object that lives in the fourth dimension, one of the most complex regular geometric objects in nature. It is bafflingly unfamiliar to nearly all human beings, because none of us has ever had the sensory experience of handling a four-dimensional object, and the only way we can visualize one is in our imagination. Even the illustrations in the article (which are excellent, by the way, and almost all the work of other Wikipedia editors than myself) are very hard to understand, because they are only illustrations of three-dimensional shadows and slices of the 600-cell, which is a four-dimensional object. The surface of the 600-cell is a curved three-dimensional space, the way the surface of the earth is a curved two-dimensional space, but even that is hard to illustrate or comprehend, because it is a curved non-Euclidean three-dimensional space, subtly different from the flat Euclidean three-dimensional every-day space we all live in. So to explain to the reader what he is looking at, even with these fine illustrations of parts of the thing, is quite a challenge.
If you read the 600-cell article section by section, line by line, I think you will find that you do not get far -- not through the first paragraph perhaps -- before you encounter a sentence that makes little sense to you, or a term or expression with which you are unfamiliar, even if you are a mathematician (and I am not). That is the reason for all the links and explanatory footnotes in the article. When you read a sentence and you don't quite get it, there is a footnote you can hover your mouse over, and a little post-it note pops up with more information, hopefully just what you need to have explained (if I have put the right footnote in the right place).
Where the concept or term has a Wikipedia article of its own, or a section of an article of its own, of course I use links in preference over footnotes, but there are hundreds of concepts and distinct mathematical terms required to enable a reader to visualize the fourth dimension, and not all of them have their own Wikipedia article (yet). Even where they do, an explanatory footnote may be needed in addition to the link, in order to explain the meaning of the term in this unique context; believe me, the 600-cell is really unique! Moreover, these ideas really can't be explained in isolation. It happens that the 600-cell is the archetype of many of those concepts, one of the few examples and in some cases the best example of a known object which has those properties. The only way I have been able to learn to visualize the fourth dimension at all is by studying the regular 4-polytopes as examples of four dimensional objects. So a quality article about the 600-cell is really an article explaining the fourth dimension, by way of the 600-cell as an example. An article which merely lists the 600-cell's properties and provides beautiful incomprehensible pictures of it would not be a high quality article. That's what we'd have if we deleted most of the footnotes, or drastically simplified them.
For the same reason that the 600-cell has a very complex geometry, the footnotes themselves have a very complex geometry. Those unfamiliar terms and strange concepts occur again and again in the article in distinct but related contexts. Whenever I write a note explaining a concept, I look for other places where the concept arises, such as in other notes about related concepts, and I add a reference to the footnote in those places, too. Just as an article may have many links to it, an explanatory note is a mini-article which may have many references to it. I hope that I have linked these notes to each other in the proper way, so that the relationships among the concepts and geometric objects is mirrored in the topology of the notes themselves, and the notes can explain the relationships among themselves to the reader. In this case, the body of footnotes itself is a polytope. Not a regular polytope to be sure, and probably of more than four dimensions in some places, but a complex geometic object to be explored. It is entirely up to the reader how to do that exploring. It is up to him whether he clicks on the footnote-within-a-footnote, just as it is up to him whether he clicks on a footnote in the first place. But if he needs to, he can explore the concept the footnote is about in depth, reviewing several notes-within-notes before he satisfies himself that he understands the sentence of the article he was reading that stumped him.
So I invite you to do more reviewing of the article and its footnotes, in greater depth if you have the interest and the time, and to tell me which footnotes were helpful to you, which were confusing or unhelpful, and which were unnecessary or redundant or badly linked. I want to improve them, and that always means simplifying them, making them more concise and briefer, wherever I can find a way to do so. You can help me to do that. Tell me if the fourth dimension makes sense to you, with or without the footnotes. Tell me where you need another footnote to explain something that is not yet explained!
Dc.samizdat (talk) 09:56, 21 March 2024 (UTC)
A symmetry operation is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections).[w] Each rotation is equivalent to two reflections, in a distinct pair of non-parallel mirror planes.[t]
Pictured are sets of disjoint great circle polygons, each in a distinct central plane of the 600-cell. For example, 4{30/3}=12{10} is an orthogonal projection of the 600-cell picturing 3 of its 72 great decagon planes, each of which can be seen as 4 disjoint great decagons.[y] The 12 great decagons lie Clifford parallel to the projection plane and to each other, and collectively constitute a discrete Hopf fibration of 12 non-intersecting great circles which visit all 120 vertices just once.
Each row of the table describes a class of distinct rotations. Each rotation class takes the left planes pictured to the corresponding right planes pictured.[aa] The vertices of the moving planes move in parallel along the polygonal isocline paths pictured.
For example, the rotation class consists of [32] distinct rotational displacements by an arc-distance of 2𝝅/3 = 120° between 16 great hexagon planes represented by quaternion group and a corresponding set of 16 great hexagon planes represented by quaternion group .[ad] One of the [32] distinct rotations of this class moves the representative vertex coordinate to the vertex coordinate .[ae]
For example, the rotation class consists of [144] distinct rotational displacements by an arc-distance of 𝝅/5 = 36° between 72 great decagon planes represented by quaternion group and a corresponding set of 72 great decagon planes represented by quaternion group .[ad] One of the [144] distinct rotations of this class moves the representative vertex coordinate to the vertex coordinate .[ae]
In a rotation class each quaternion group may be representative not only of its own fibration of Clifford parallel planes[ad] but also of the other congruent fibrations.[y] For example, rotation class takes the 4 hexagon planes of to the 4 hexagon planes of which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,[av] all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes , and . The name is the conventional representation for all [16] congruent plane displacements.
These rotation classes are all subclasses of which has [32] distinct rotational displacements rather than [16] because there are two chiral ways to perform any class of rotations, designated its left rotations and its right rotations. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.[ax] Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.
Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,[z] through a characteristic rotation angle.[ac] For example, the rotation moves all [16] hexagonal planes at once by 2𝝅/3 = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same orientation, passing through all 4 planes of the left set and all 4 planes of the right set once each.[ab] The picture in the isocline column represents this union of the left and right plane sets. In the example it can be seen as a set of 4 Clifford parallel skew hexagrams, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.[aw]
Triacontagram {30/12}=6{5/2} is the Schläfli double six configuration 125302 characteristic of the H4 polytopes.[bb] The 30 vertex circumference is the Petrie polygon.[bc] The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.[ac] In the 120-cell, the six disjoint pentagrams are inscribed regular 5-cells of edge-length √5/2.
Twelve more years passed.... Bilbo was going to be eleventy-one, 111, a rather curious number ... and Frodo was going to be thirty-three, 33, the date of his ‛coming of age’.[bj]
The circumference of a 30-cell ring varies from 10 edge lengths on a decagon great circle, to 30 edge lengths along its helical Petrie polygon. The circumference of an isocline along its chords (which are 24-cell edges) is 15. The shortest path around a 30-cell ring on 600-cell edges is 20 edge-lengths
Each vertex of the 30-cell ring is 20 edge lengths distant from itself around the isocline by the shortest path along 600-cell edges, which is a strip of triangles edge-bonded at those 20 edges; but it is 21 tetrahedra distant from itself by the shortest chain of edge-bonded tetrahedra. Perhaps this difference of 1 edge length occurs for the same reason that Phineas Fogg passed through 81 distinct days (sunrise-sunset pairs) on his journey Around the World in 80 Days. Phineas Fogg performed one slow forward somersault in the course of his 80 day journey, in addition to the 80 times he was rotated all the way around the planet. Each day he hastened to his sunset a little faster than the planet would normally have brought him to it, and each of his days was a little shorter than 24 hours. He rotated through 81 distinct "solar" rotations (which of course are not really rotations of the sun, but rotations of ourselves about the earth with respect to the sun). We can see his extra day as a consequence of his slow extra somersault, as he himself was rotated in place once in the course of his 80-rotation journey. In a single 360° isoclinic rotation of the 600-cell, each of its 600 tetrahedral "passengers" is itself rotated in place once, in addition to being rotated all the way around the 600-cell. Each tetrahedron is rotated in the first place at each of the 10 36°x36° steps of the rotation to another tetrahedron two edge-lengths away, in a different rotational orientation.
Of course there is also a unit-radius great circle in this central plane (completely orthogonal to a decagon central plane), but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects none of the points of the 600-cell.
Any five 24-cells which can be reached by either 5-click simple rotation by itself have disjoint vertices.
We can give the 25 24-cells physical addresses of the form (i, j) corresponding to their inclination from an origin 24-cell,[bn] where i and j are integers between 0 and 4 corresponding to multiples of 𝜋/5 in the two orthogonal planes.[bo] The 25 overlapping 24-cells are thus arranged logically by address (not physically) in a 5 x 5 array.[bp] The five 24-cells in each row of this array (any five 24-cells with the same i) are disjoint: they have disjoint sets of vertices that together account for all 120 vertices of the 600-cell. The five 24-cells in each column of the array (any five with the same j) are similarly disjoint. No other set of five 24-cells is disjoint; these are the only ten ways to partition the 25 24-cells into five disjoint 24-cells.
Notice that the 600-cell has two pentagons inscribed in each decagon (as the 24-cell has two triangles inscribed in each hexagon). The pentagon's √1.𝚫 edge chord falls between the √1 hexagon edge chord and the √2 square edge chord in length. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges), which has a short radius between those of the 24-cells' envelopes of octahedra (made of √1 hexagon edges) and the 16-cells' envelopes of tetrahedra (made of √2 square edges). Consider also the √2.𝚽 = φ and √3.𝚽 chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell.[bq] The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell; it also inserts new features deep in the interstices of the 24-cell's interior structure, which it inherits in full, compounds five-fold, and then elaborates on.
Euler's identity has a geometric representation as a rotation in the complex plane. Euler's identity can be interpreted as saying that rotating any point radians around the origin has the same effect as reflecting the point across the origin. Similarly, the derivative equation can be interpreted as saying that rotating any point by one turn around the origin (in 2-dimensional space) returns it to its original position.
Euler's identity captures the relation between the Coxeter group operations rotation and reflection, in Euclidean spaces of any number of dimensions.[t]
Like all regular convex polytopes, the dodecahedron can be dissected into an integral number of disjoint orthoschemes, all of the same shape characteristic of the polytope. A polytope's characteristic orthoscheme is a fundamental property because the polytope is generated by reflections in the facets of its orthoscheme. The orthoscheme occurs in two chiral forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a quadrirectangular irregular tetrahedron.
The faces of the dodecahedron's characteristic tetrahedron lie in the dodecahedron's mirror planes of symmetry. The dodecahedron's symmetry group is denoted Ih or H3. The dodecahedron and its dual polytope, the icosahedron, have the same symmetry group but different characteristic tetrahedra.
The characteristic tetrahedron of the regular dodecahedron can be found by a canonical dissection[15] of the regular dodecahedron which subdivides it into 120 of these characteristic orthoschemes surrounding the dodecahedron's center. Five left-handed orthoschemes and five right-handed orthoschemes meet in each of the dodecahedron's twelve faces, the ten orthoschemes collectively forming a pentagonal pyramid with the dodecahedron's pentagonal face as its equilateral base, and its apex at the center of the dodecahedron.
If the dodecahedron has radius 𝒍 = 2, its characteristic tetrahedron's six edges have lengths , , around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[ch] plus , , (edges that are the characteristic radii of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is , , , first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 90-60-30 triangle which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a 45-90-45 triangle with edges , , , a right triangle with edges , , , and a right triangle with edges , , .
Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.[e]
The characteristic 5-cell of the regular 600-cell is represented by the Coxeter-Dynkin diagram, which can be read as a list of the dihedral angles between its mirror facets.
It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron.
The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.
The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).[k] If the regular 600-cell has unit radius and edge length , its characteristic 5-cell's ten edges have lengths , , around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[ch] plus , , (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus , , , (edges which are the characteristic radii of the 600-cell).
The 4-edge path along orthogonal edges of the orthoscheme is , , , , first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.
Articles I write or contribute to which contain some original research, and so cannot be published as Wikipedia articles, are hosted at Wikiversity instead.
Koca's slides from a Bangalore conference summarizing his studies of uniform 4-polyopes; he has described quite a few hitherto-nondescript 4-polytopes, such as the 720-point 720-cell (600 octahedra and 120 isosahedra), and many nondescript duals of known uniform 4-polytopes; some of these polytopes have vertices on two different concentric 3-spheres, so they reveal relationships between 4-polytopes of different radii
https://slideplayer.com/slide/8639634/
As the invariant axis of a rotating 2-sphere is dimensionally analagous to the invariant plane of a rotating 3-sphere, the poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle. The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,[cr] but also completely orthogonal. The invariant great circles of the 4D rotation are its poles. In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on one such circle (never on two, since the completely orthogonal circles, like all the Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, an isoclinic 4D rotation of the 3-sphere has nothing but poles, an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles. In one full revolution of such a rotation, every point in the space loops exactly once through one of these pole-circles.[cs] The circles are arranged with a surprising symmetry, so that each pole-circle links with every other pole-circle, like an infinitely dense 4D fabric of chain mail, and no 2 circles ever intersect.
^Cite error: The named reference missing the nearest vertices was invoked but never defined (see the help page).
^ abCite error: The named reference pairs of completely orthogonal planes was invoked but never defined (see the help page).
^√3/4 ≈ 0.866 is also the long radius of the √2-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal (they are inclined to each other at 120°), and they radiate symmetrically compressed into 3 dimensions not 4. Conversely, in the 4 dimensional 24-cell isoclinic rotation the four √3/4 ≈ 0.866 displacements summing to a 120° degree displacement are not radii, but rather successive orthogonal steps in a path extending in 4 dimensions not 3, along the orthogonal edges of a 4-orthoscheme. √3/4 ≈ 0.866 is also the area of the equilateral triangle face of both the 3-dimensional unit-edge tetrahedron and the 4-dimensional unit-edge, unit-radius 24-cell.
^In an isoclinic rotation, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a 4-dimensional diagonal. The point is displaced a total Pythagorean distance equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.[a] For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,[b] each vertex is displaced to another vertex √3 (120°) away, moving √3/4 ≈ 0.866 in four orthogonal directions.[c]
^ abAn orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own facets (its mirror walls).
Every regular polytope can be dissected radially into instances of its characteristic orthoscheme surrounding its center.
The characteristic orthoscheme has the shape described by the same Coxeter-Dynkin diagram as the regular polytope without the generating point ring.
^In addition to four hexagonal central planes (inclined at 60° to each other), the cuboctahedron also has planes of symmetry that bisect its faces.
^ abcdThe 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated , , and .[ab] Each named set of 4 Clifford parallel[i] hexagons comprises a discrete fibration covering all 24 vertices. Cite error: The named reference "four hexagonal fibrations" was defined multiple times with different content (see the help page).
^In this orthogonal projection of the 24-cell to an {8/4}=4{2} octagram, each point represents 3 vertices and each line represents multiple 24-cell axes. The 4 lines can be seen as four 24-cell axes inclined to each other at 60° isoclinically: not the four orthogonal axes of a 16-cell, but 4 Clifford parallel great digon planes.[b] The 4 great digon planes of lie completely orthogonal to a fibration of 4 great hexagon planes also designated.[g] Note that the 4 great digon planes do not constitute a complete fibration of 24 points, and that the symbol represents both a complete fibration of 4 Clifford parallel great hexagon planes and the set of 4 great digon planes completely orthogonal to it.
^ abcdCite error: The named reference Clifford parallels was invoked but never defined (see the help page).
^ abcCite error: The named reference isoclinic geodesic was invoked but never defined (see the help page).
^ abThe four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius.
The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center.
Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme.
The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
^Cite error: The named reference common core was invoked but never defined (see the help page).
^The smallest of the four nested 24-cells is the outer 24-cell's insphere-inscribed dual formed by joining its cell centers, which is the common core of the 24-cell and its inscribed 8-cells and 16-cells.[l]
^The track has five Clifford parallel[i] rails (one carrying each vertex of the moving 5-cell) that spiral around each other in a quintuple helix, each parallel strand of which is a doubly-circular isocline, not a simple great circle.[j] Although the train cars are alternating left-hand and right-hand 4-orthoschemes, all five parallel rails wind always in the same direction (twice around the 24-cell), leftward or rightward as the isoclinic rotation is left or right. Each vertex of the moving 4-orthoscheme follows a different geodesic isocline (a different rail that the 5-wheeled train cars run on) along a 12-vertex loop that repeats after two revolutions. However, only one of the five rails runs through the 24-cell's vertices, because only one vertex of each 4-orthoscheme is a vertex of the 24-cell.[k] The five Clifford parallel rails carrying the five vertices of the moving 5-cell are geodesic isoclines of five different radii, only one of which lies on the same 3-sphere as the 24-cell; the other four rails each visit 12 points on the surface of a smaller 3-sphere on which (respectively) the edge centers, face centers, cell centers, or the 24-cell center lie. (The smallest 3-sphere is the 24-cell center itself, of radius 0, and its "rail" has 12 vertices which are all the same stationary point.) We can visualize these five concentric 3-spheres as four dual 24-cells nested around their common center.[m] Thus each rail does run through a set of 24-cell vertices, but in a 24-cell of its own characteristic radius.
^96 such 720° rings of 12 characteristic 5-cells (with disjoint 5-cells) spiral around each other, pass through each other, and nest together (sharing tetrahedral cells) to entirely fill the 24-cell with their 1152 5-cells.
^In an isoclinic rotation each moving 4-orthoscheme as a set of 5 vertices returns to its original set of vertices after two revolutions, but in a different orientation entanglement. When after 12 reflections the 5-cell arrives back at its original position, each doubly-curved rail[j] is continuous with the rail it originally departed on, but the more-than-doubly-curved train of 5-cells is no longer reflecting forward in the same direction in which it originally departed. Therefore if it continues in its orbit, it will not immediately revisit the same sequence of 12 stationary 5-cells. The 5-cell returns in its original orientation only after 192 revolutions (a 384𝜋 orbit), after passing through all 1152 stationary 5-cells that fill the 24-cell. The moving 5-cell is rotating as it revolves, and in each of the 1152 5-cells it visits in a 384𝜋 orbit it has a unique orientation.
^ abThe hexagonal planes in the 600-cell occur in equi-isoclinic[bf] groups of 4, everywhere 4 Clifford parallel hexagons 60° (𝝅/3) apart form a 24-cell.
Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° (𝝅/5) apart, 4 72° (2𝝅/5) apart, 4 108° (3𝝅/5) apart, and 4 144° (4𝝅/5) apart, for a total of 20 Clifford parallel hexagons (120 vertices) that comprise a discrete Hopf fibration.
^ abCite error: The named reference double rotation was invoked but never defined (see the help page).
^ abcCayley showed that any rotation in 4-space can be decomposed into two isoclinic rotations,[r] which intuitively we might see follows from the fact that a transformation from one inertial reference frame to another is itself just a rotation in 4-dimensional Euclidean space. Cite error: The named reference "Cayley's rotation factorization into two isoclinic reference frame transformations" was defined multiple times with different content (see the help page).
^ abcLet Q denote a rotation, R a reflection, T a translation, and let Qq Rr T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q2 is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
Qq Rr
where 2q + r ≤ n, the number of dimensions. Transformations involving a translation are expressible as
Qq Rr T
where 2q + r + 1 ≤ n.
For n = 4 in particular, every displacement is either a double rotation Q2, or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the principle of relativity, every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q2 in a linearly moving (translating) reference frame. By the same principle, we can view any QT or Q2 as an isoclinic (equi-angled) Q2 by appropriate choice of reference frame.[s]] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.[2]Cite error: The named reference "transformations" was defined multiple times with different content (see the help page).
It can be found in a regular decagon by connecting any two adjacent vertices to the center, and in the regular pentagon by connecting any two adjacent vertices to the vertex opposite them.
The vertex angle is:
𝛉 = arccos(φ/2) = 𝜋/5 = 36°
so the base angles are each 2𝜋/5 = 72°.
The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.
^The long radius (center to vertex) of the 600-cell is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1/φ if its radius is 1.
Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon.
(The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.)
Radially golden polytopes are those which can be constructed, with their radii, from golden triangles[u] which meet at the center, each contributing two radii and an edge.
^ abcThe F4 regular polytope (the 24-cell) has 576 distinct simple rotations, each in its invariant rotation plane. The 576 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of .. distinct classes of isoclinic rotations, and are usually given as those sets.[11]Cite error: The named reference "distinct rotations" was defined multiple times with different content (see the help page).
^Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for one of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of one rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.
^ abCite error: The named reference six decagonal fibrations was invoked but never defined (see the help page).
^ abIn an isoclinic rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a simple rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.
^The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,[z] but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.
^ abcThe and sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.
^ abcdTwo angles are required to fix the relative positions of two planes in 4-space.[5]
Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space.
Great decagons are a multiple (from 0 to 4) of 36° (𝝅/5) apart in each angle, and may be the same angle apart in both angles.[bh]
Great hexagons may be 60° (𝝅/3) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° (𝝅/5) apart in one or both angles.[q]
Great squares may be 90° (𝝅/2) apart in one or both angles, may be 60° (𝝅/3) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° (𝝅/5) apart in one or both angles.[bi]
Planes which are separated by two equal angles are called isoclinic.[bf]
Planes which are isoclinic have Clifford parallel great circles.[i]
A great hexagon and a great decagon are neither isoclinic nor Clifford parallel; they are separated by a 𝝅/3 (60°) angle and a multiple (from 1 to 4) of 𝝅/5 (36°) angle.
^ abcdA quaternion group corresponds to a distinct set of Clifford parallel great circle polygons, e.g. corresponds to a set of four disjoint great hexagons.[g] Note that and generally are distinct sets. The corresponding vertices of the planes and the planes are 180° apart.[ac]Cite error: The named reference "quaternion group" was defined multiple times with different content (see the help page).
^ abcdeA 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 standard (vertex-up) orientation is , the Cartesian "north pole". Thus e.g. designates a √1 chord of 60° arc-length. Each such distinct chord is an edge of a distinct great circle polygon, in this example a great hexagon, intersecting the north and south poles. 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 north and south poles. This quaternion coordinate is thus representative of the 4 disjoint great hexagons pictured, a quaternion group[ad] which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.[g]Cite error: The named reference "north pole relative coordinate" was defined multiple times with different content (see the help page).
^In this orthogonal projection of the 600-cell to a {30/10}=10{3} triacontagram, each point represents four vertices, each line represents multiple √3 chords, and each triangle represents multiple {6/2} hexagram isoclines. Each triangle can be seen as two superimposed open skew triangles, with their opposite ends joined to form a hexagram winding through all four orthgonal dimensions, such that it appears to be a triangle in this projection to two dimensions. The 10 triangles can be seen as a fibration of [20] Clifford parallel hexagram isoclines.
^The edges and 8𝝅 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 5𝝅 characteristic rotations of the 600-cell lie in the great pentagon ✩ (great decagon) central planes. Rotations of this type are an expression of the symmetry group . The edges and 4𝝅 characteristic rotations of the other regular 4-polytopes, the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell, all lie in the great triangle △ (great hexagon) central planes. Collectively these rotations are expressions of all four symmetry groups , , and .
^In this orthogonal projection of the 600-cell to a {30/10}=10{3} triacontagram, each point represents four vertices, each line represents multiple √3 chords, and each triangle represents multiple {6/2} hexagram isoclines. Each triangle can be seen as two superimposed open skew triangles, with their opposite ends joined to form a hexagram winding through all four orthgonal dimensions, such that it appears to be a triangle in this projection to two dimensions. The 10 triangles can be seen as a fibration of [20] Clifford parallel hexagram isoclines.
^Cite error: The named reference hybrid rotation was invoked but never defined (see the help page).
^Cite error: The named reference invariant planes of an isoclinic rotation was invoked but never defined (see the help page).
^ abCite error: The named reference clasped hands was invoked but never defined (see the help page).
^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.[aw]
^Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders in order to move the short distance between Clifford parallel subspaces.
^Cite error: The named reference isoclinic chessboard was invoked but never defined (see the help page).
^In the same sense that the 3-sphere is the product of two completely orthogonal cylinders, each cell ring can be seen to be twisting in two completely orthogonal cylindrical spirals at once.[ay]
In one circular cylindrical spiral its pentadecagram2 5𝝅 isoclines wind twice around the circumference of the 30-cell ring (in a 720° rotation) visiting their 15 vertices in two revolutions.
The disjoint orange and yellow 15-grams each visit 15 of the 30 vertices, which alternate colors like the black and white squares of a chessboard.[az]
(The isoclinic rotation as a whole visits 60 of the 120 vertices on the first revolution, and their antipodal 60 vertices on the second.)
From the orthogonal viewpoint, looking down the axis of the 30-cell ring cylinder, the 5𝝅 isocline can be seen winding 6 times around the 600-cell in the same 720° rotation, circling through 3 concatenated open-ended pentagram2 helices linked end-to-end.[4]
^ abThe Schläfli double six configuration 125302 is realized in the 600-cell's six decagonal fibrations.
Each Hopf fibration comprising the 600-cell is a bundle of 12 disjoint (Clifford parallel) decagon great circle fibers (12 "lines") and 30 disjoint sets of 4 vertices (30 "points"). The fibration has 20 30-cell rings, in 5 sets of 4 disjoint 30-cell rings.
, each of which has 5 "intersection points" with the fibration's four disjoint 30-cell rings of regular tetrahedra. Each "intersection point" is a tetrahedral cell between the four disjoint 30-cell rings. The "points" of the configuration are tetrahedral cells which have one vertex in each of the four disjoint 30-cell rings, bridging the four rings; these tetrahedra are not cells in any of those four 30-cell rings.
The 12 decagons ("lines") can be partitioned into two subsets of six decagons: each decagon is disjoint from (Clifford parallel to) the decagons in its own subset of six, and intersects all but one of the decagons in the other subset of six lines. Each of the 12 lines of the configuration contains five intersection points, and each of these 30 intersection points belongs to exactly two lines, one from each subset,
The 12 great decagon "lines" can be partitioned into two subsets of 6 great pentagons (with 2 pentagons inscribed in each decagon); each pentagon is Clifford parallel to (completely disjoint from) the pentagons of its own subset of 6 great circles, and intersects all but one of the pentagons of the other subset of 6 great circles (the one inscribed in the same decagon).
In the 600-cell the "double sixes" are great hexagons {6} in unique 24-cells {3,4,3}.[bd]
The 30-cell ring's 2 axial pentadecagram isoclines together contain edges from 5 hexagons in 5 different 24-cells; each hexagon is the intersection of exactly 2 different 24-cells, and the 2 isoclines.
Each of the 30-cell ring's 3 great decagons contains two inscribed pentagons {5}.
Each pentagon contains one vertex of each of the 5 hexagons in 5 completely disjoint 24-cells; each {5} is a ring linking 5 completely disjoint 24-cells.[be]
Each hexagon contains one vertex of each of the 6 pentagons in 4 completely disjoint 30-cell rings; each {6} is a ring linking 4 completely disjoint 30-cell rings.
Together the 2 isoclines axial to each 30-cell ring intersect all 30 vertices of the ring, and all 6 pentagons inscribed in the 3 (of 12) Clifford parallel great decagons which bound each 30-cell ring.
^ abcThe regular skew 30-gon is the Petrie polygon of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell Boerdijk–Coxeter helix rings: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell).
^Cite error: The named reference disjoint from 8 and intersects 16 was invoked but never defined (see the help page).
^Cite error: The named reference 24-cells bound by pentagonal fibers was invoked but never defined (see the help page).
^ abcdIn 4-space no more than 4 great circles may be Clifford parallel[i] and all the same angular distance apart.[6]
Such central planes are mutually isoclinic: each pair of planes is separated by two equal angles, and an isoclinic rotation by that angle will bring them together.
Where three or four such planes are all separated by the same angle, they are called equi-isoclinic.
^Cite error: The named reference Boerdijk–Coxeter helix was invoked but never defined (see the help page).
^The decagonal planes in the 600-cell occur in equi-isoclinic[bf] groups of 3, everywhere 3 Clifford parallel decagons 36° (𝝅/5) apart form a 30-cell Boerdijk–Coxeter triple helix ring.[bg]
Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 12 Clifford parallel decagons (120 vertices) that comprise a discrete Hopf fibration.
Because the great decagons lie in isoclinic planes separated by two equal angles, their corresponding vertices are separated by a combined vector relative to both angles.
Vectors in 4-space may be combined by quaternionic multiplication, discovered by Hamilton.[7]
The corresponding vertices of two great polygons which are 36° (𝝅/5) apart by isoclinic rotation are 60° (𝝅/3) apart in 4-space.
The corresponding vertices of two great polygons which are 108° (3𝝅/5) apart by isoclinic rotation are also 60° (𝝅/3) apart in 4-space.
The corresponding vertices of two great polygons which are 72° (2𝝅/5) apart by isoclinic rotation are 120° (2𝝅/3) apart in 4-space, and the corresponding vertices of two great polygons which are 144° (4𝝅/5) apart by isoclinic rotation are also 120° (2𝝅/3) apart in 4-space.
^The square planes in the 600-cell occur in an equi-isoclinic[bf] group of 2, everywhere 2 Clifford parallel squares 90° (𝝅/2) apart form a 16-cell.
Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° (𝝅/3) apart form a 24-cell.
Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° (𝝅/5) apart, 3 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 30 Clifford parallel squares (120 vertices) that comprise a discrete Hopf fibration.
^When Bilbo was 99, Frodo was 21. 12 years later Bilbo was 111 and Frodo was 33. Bilbo and Frodo both left home on their respective adventures at the age of 50. When Bilbo left home, Frodo's birth was still 21 years in the future, when Bilbo would be 71. When Frodo left home, 17 years after Bilbo's 111th birthday party, Bilbo was 128.
^Cite error: The named reference completely orthogonal Clifford parallels are special was invoked but never defined (see the help page).
^ abIn the 16-cell each single isocline winds through an entire fibration of two completely orthogonal great squares (all 8 vertices).[bk] Two fibers which intersect cannot be a fibration. We may consider each left or right 16-cell isocline to be a whole fibration by itself, consisting of a single fiber. The 16-cell is the only place where such a discrete fibration of one isocline fiber occurs.[bm]
^Except in the 16-cell,[bl] a pair of left and right isocline circles[j] have disjoint vertices and belong to the same fibration. The left and right isocline helices are Clifford parallel (non-intersecting) but counter-rotating, forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect).
^The origin 24-cell can be chosen arbitrarily (it can be any of the 25). Without loss of generality we can choose the 24-cell whose coordinates are the permutations of (0, 0, 0, ±1) and (±1/2, ±1/2, ±1/2, ±1/2).
^In 4 dimensions, inclination is 2-dimensional, because two angles are required to characterize the inclination of one object with respect to another in a 4-dimensional space. 4 dimensional rotations in general are fully characterized only as a pair of simple rotations around two orthogonal planes (a double rotation).
^The rotational orientations of the 25 24-cells within the 600-cell, expressed as inclinations (double rotation angles of arc in units of 𝜋/5) from the origin 24-cell (0,0):
Each row and each column is a set of 5 24-cells with disjoint vertices, accounting for all 120 vertices of the 600-cell.[9]
^The √2.𝚽 = φ and √3.𝚽 chords produce irregular interior faces and cells, since they make isosceles great circle triangles out of two chords of their own size and one of another size.
^Cite error: The named reference isoclinic rotation was invoked but never defined (see the help page).
^ abcIn symmetry group F4 the operations [32]𝑹q7,q7 and [32]𝑹q7,-q7 are the 64 distinct simple rotations which comprise the characteristic hexagonal rotations of the 24-cell, in 16 regular great hexagons with √1 edges. Mamone's[12]Cite error: The named reference "#5 edge rotation" was defined multiple times with different content (see the help page).
^ abCite error: The named reference hexagram was invoked but never defined (see the help page).
^The preceding this quaternion coordinate indicates that this entire row of the table is to be duplicated, with the positive and negative coordinate value respectively appearing here. Thus this row of the table represents two distinct pairings of isocline chords, and twice as many invariant great circle polygons as indicated in the leftmost column.
^ abCite error: The named reference 4𝝅 rotation was invoked but never defined (see the help page).
^ abCite error: The named reference 120-cell characteristic rotation was invoked but never defined (see the help page).
^In symmetry group 𝛨4 the operation [1200]𝑹7,3 is the 1200 distinct simple rotations which comprise the characteristic pentadecagram isoclinic rotation of the 120-cell, in 200 invariant planes containing irregular great dodecagons with #4 and #1 edges.[bw] The left and right isoclines consist of 15 #7 chords and 15 #3 chords, respectively. [3]
^The preceding this quaternion coordinate indicates that this entire row of the table is to be duplicated, with the positive and negative coordinate value respectively appearing here. Thus this row of the table represents two distinct pairings of isocline chords, and twice as many invariant great circle polygons as indicated in the leftmost column.
^ abcCite error: The named reference 120-cell Petrie {30}-gon was invoked but never defined (see the help page).
^The 120-cell contains central planes intersecting {4} vertices in a √0.5・√3.5 great rectangle. It also contains central planes intersecting {12} vertices in an irregular great dodecagon, which contains various inscribed {3} and {6} great polygons. The √0.5 and √3.5 chords are bridges between two irregular great dodecagon {12} planes which the great rectangle intersects in its two √4 diameters.
^The 56°・124° complementary pair of chords occupy their own {4} vertex ☐ rectangular great circle plane, which intersects (in its two √4 axes) the other two kinds of great circle plane, the {12} vertex △ great hexagon / great triangle plane and the {10} vertex ✩ great decagon / great pentagon plane. The 56° and 124° chords are bridges between a {12} plane and a {10} plane.
^The point itself (𝜉i, 𝜂, 𝜉j) does not necessarily lie in either of the invariant planes of rotation referenced to locate it (by convention, the wz and xy Cartesian planes), and never lies in both of them, since completely orthogonal planes do not intersect at any point except their common center. When 𝜂 = 0, the point lies in the 𝜉i "longitudinal" wz plane; when 𝜂 = 𝜋/2 the point lies in the 𝜉j "equatorial" xy plane; and when 0 < 𝜂 < 𝜋/2 the point does not lie in either invariant plane. Thus the 𝜉i and 𝜉j coordinates number vertices of two completely orthogonal great circle polygons which do not intersect (at the point or anywhere else).
^The angles 𝜉i and 𝜉j are angles of rotation in the two completely orthogonal invariant planes[cc] which characterize rotations in 4-dimensional Euclidean space. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to 𝜋/2. The (𝜉i, 0, 𝜉j) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉i, 𝜋/2, 𝜉j) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉i, 0 < 𝜂 < 𝜋/2, 𝜉j) describe the great circles (not "lines of latitude") which cross an equator but do not pass through the north or south pole.
^ abThe conversion from Hopf coordinates (𝜉i, 𝜂, 𝜉j) to unit-radius Cartesian coordinates (w, x, y, z) is:
w = cos 𝜉i sin 𝜂
x = cos 𝜉j cos 𝜂
y = sin 𝜉j cos 𝜂
z = sin 𝜉i sin 𝜂
The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).
The "Cartesian north pole" (1, 0, 0, 0) is Hopf (0, 𝜋/2, 0).
^The Hopf coordinates (also known as the toroidal coordinates of S3) are triples of three angles:
(𝜉i, 𝜂, 𝜉j)
that parameterize the 3-sphere by numbering points along its great circles.[14] A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).[cd] The 𝜉i and 𝜉j coordinates range over the vertices of completely orthogonal great circle polygons which do not intersect at any vertices. Hopf coordinates are a natural alternative to Cartesian coordinates[ce] for framing regular convex 4-polytopes, because the group of rotations in 4-dimensional Euclidean space, denoted SO(4), generates those polytopes. A rotation in 4D of a point {ξi, η, ξj} through angles ξ1 and ξ2 is simply expressed in Hopf coordinates as {ξi + ξ1, η, ξj + ξ2}.
where {<k} is the {permutation} of the k non-negative integers less than k, and {≤k} is the permutation of the k+1 non-negative integers less than or equal to k. Each coordinate permutes one set of the 4-polytope's great circle polygons, so the permuted coordinate set expresses one set of rotations in 4-space which generates the 4-polytope. With Cartesian coordinates the choice of radius is a parameter determining the reference frame, but Hopf coordinates are radius-independent: all Hopf coordinates convert to unit-radius Cartesian coordinates by the same mapping. [ce] Unlike Cartesian coordinates, Hopf coordinates are not necessarily unique to each point; there may be Hopf coordinate synonyms for a vertex. The multiplicity 𝑚 of the coordinate permutation is the ratio of the number of Hopf coordinates to the number of vertices.
^ abcd(Coxeter 1973) harv error: no target: CITEREFCoxeter1973 (help) uses the greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
^In the dodecahedral cell of the unit-radius 120-cell, the length of the edge (the #1 chord of the 120-cell) is 1/φ2√2 ≈ 0.270. The orange vertices lie at the Cartesian coordinates (±φ3√8, ±φ3√8, ±φ3√8) relative to origin at the cell center. They form a cube (dashed lines) of edge length 1/φ√2 ≈ 0.437 (the pentagon diagonal, and the #2 chord of the 120-cell). The face diagonals of the cube (not shown) of edge length 1/φ ≈ 0.618 are the edges of tetrahedral cells inscribed in the cube (600-cell edges, and the #3 chord of the 120-cell). The diameter of the dodecahedron is √3/φ√2 ≈ 0.757 (the cube diagonal, and the #4 chord of the 120-cell).
^ abInscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,[20] of edge-length √5/2. No unit-radius regular 4-polytope except the 5-cell and the 120-cell contains √5/2 chords. The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells three different ways. Each √5/2 chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells. These chords and the 120-cell edges are the only chords between vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and √5/2 apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells (three different ways). Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together in a distinct isoclinic rotation.
^The black and white pentadecagram isoclines both act as either a right or a left isocline in the distinct right or left isoclinic rotation.
^Cite error: The named reference non-planar geodesic circle along edges was invoked but never defined (see the help page).
^The 120 regular 5-cells are completely disjoint; the vertices of two 5-cells are linked only by 120-cell edges, not by 5-cell edges or any other chords. Therefore each pentadecagram is confined to a single 5-cell. Each pentadecagram is a circular path along 15 edges belonging to the same 10-edge 5-cell. Each 5-cell contains three distinct Petrie pentagons of its 5 vertices, pentagonal circuits each binding 5 disjoint 600-cells together. Additionally it has two distinct pentadecagram circuits, each of which visits each vertex three times. Each pentadecagram isocline is a distinct sequence of the 15 edges of all three distinct Petrie pentagons, but it does not include any of those 5-circuits as subsequences; it is not the concatenation of the three 5-circuits, but their interleaving. Successive vertices of each pentadecagram are vertices in completely disjoint 600-cells, as are the 5 vertices of the 5-cell. Notice that the pentadecagram isocline and its distinct isoclinic rotation is a property of the individual 5-cell, independent of the existence of a 120-cell.[3]
^In triacontagram {30/8}=2{15/4}, 2 disjoint pentadecagram isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.[ck] The chords join vertices which are 8 120-cell edges apart on the zig-zag Petrie polygon (not shown) which joins the 30 vertices of the circumference.[bc] The characteristic isoclinic rotation of the 120-cell takes place in the invariant planes of its 1200 edges and the completely orthogonal invariant planes of its inscribed regular 5-cells' edges. There is one distinct characteristic right (left) isoclinic rotation. The rotation's isocline chords of length √5/2 are the 1200 edges of 120 disjoint regular 5-cells inscribed in the 120-cell.[cj] 15 chords join vertices 8 edge-lengths apart in a geodesic pentadecagram circle.[cl] Successive chords of each pentadecagram are edges of the same regular 5-cell.[cm]
^Let Q denote a rotation, R a reflection, T a translation, and let Qq Rr T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q2 is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
Qq Rr
where 2q + r ≤ n, the number of dimensions.[21]
^ abThe 120-cell has 600 vertices distributed symmetrically on the surface of a [3-sphere] in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 rays [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a basis. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays.[22]
^The 120-cell can be constructed as a compound of 5 disjoint 600-cells, or 25 disjoint 24-cells, or 75 disjoint 16-cells, or 120 disjoint 5-cells. Except in the case of the 120 5-cells, these are not counts of all the distinct 4-polytopes which can be found inscribed in the 120-cell, only the counts of completely disjoint inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains 10 distinct 600-cells, 225 distinct 24-cells, and 675 distinct 16-cells.[cp]
^Cite error: The named reference Hopf fibration base was invoked but never defined (see the help page).
^Consider the truth of this statement. It can be found in the literature, expressed in the mathematical language of the Hopf fibration,[32] but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting in the completely orthogonal plane. With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle. Each helical isocline is itself a kind of circle, but it is not a planar great circle of the Hopf fibration: it is a special kind of geodesic circle whose circumference is greater than 2𝝅r, and it is not pictured explicitly at all by the plain statement we are trying to visualize. So we cannot visualize this particular statement about the Hopf great circles in a stationary reference frame. The statement does not simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle as every great circle itself is moving orthogonally, flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous twisting rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different great circles, because Clifford parallel circles are not parallel in the ordinary sense, so every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once in a full isoclinic revolution, every vertex moves more than 360 degrees, as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique kind of Hopf fibration,[33] and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the characteristic isoclinic rotation of the great hexagon fibration of the 24-cell, each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle); but in the characteristic isoclinic rotation of the great decagon fibration of the 600-cell, each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).
^ abUberti, R.; Janse van Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.), Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89–100, doi:10.1007/978-1-4612-1712-1_9, MR1655039; see Table 2, p. 97
^ abcCoxeter 1973, pp. 217–218, §12.2 Congruent transformations. sfn error: no target: CITEREFCoxeter1973 (help)
^ abcdMamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations. sfn error: no target: CITEREFMamonePileioLevitt2010 (help)Cite error: The named reference "FOOTNOTEMamonePileioLevitt20101438–1439§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations" was defined multiple times with different content (see the help page).
^Sadoc 2001, pp. 576–577, §2.4: the six-fold screw axis.
^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.)" sfn error: no target: CITEREFKimRote2016 (help)
^Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors and according to sfn error: no target: CITEREFMamonePileioLevitt2010 (help)
^Coxter 1973, p. 233. sfn error: no target: CITEREFCoxter1973 (help)
^van Ittersum 2020, pp. 84–85, §4.3.3. sfn error: no target: CITEREFvan_Ittersum2020 (help)
^ abMamone, Pileio & Levitt 2010, §4.5 Regular Convex 4-Polytopes, Table 2; gives each class of distinct rotation as [n]Rql,qr where n is the count of distinct simple rotations in the class and ql and qr are the left and right isocline chords defining the distinct class, respectively. The chords are given as Cartesian coordinates (quaternions) of a representative chord of the distinct length.[ae] sfn error: no target: CITEREFMamonePileioLevitt2010 (help)Cite error: The named reference "FOOTNOTEMamonePileioLevitt2010§4.5 Regular Convex 4-Polytopes, Table 2" was defined multiple times with different content (see the help page).
^ abMamone, Pileio & Levitt 2010, pp. 1438–1439, q7 is the &radic, 1 edge of the 24-cell, -q7 is the &radic, 3 chord., §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations. sfn error: no target: CITEREFMamonePileioLevitt2010 (help)
^Zamboj 2021, pp. 10–11, §Hopf coordinates. sfn error: no target: CITEREFZamboj2021 (help)
^Sadoc 2001, pp. 575–576, §2.2 The Hopf fibration of S3.
^Coxeter 1973, p. 130, §7.6 The symmetry group of the general regular polytope; "simplicial subdivision". sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, pp. 292–293, Table I(i); "Dodecahedron". sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, pp. 292–293, Table I(ii); "600-cell". sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, p. 139, §7.9 The characteristic simplex. sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, p. 290, Table I(ii); "dihedral angles". sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, p. 304, Table VI (iv): 𝐈𝐈 = {5,3,3}. sfn error: no target: CITEREFCoxeter1973 (help)
^Coxeter 1973, p. 217, §12.2 Congruent transformations. sfn error: no target: CITEREFCoxeter1973 (help)
^Waegell & Aravind 2014, pp. 3–4, §2 Geometry of the 120-cell: rays and bases. sfn error: no target: CITEREFWaegellAravind2014 (help)
