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Literally the Greek word "isogon" means: having alike knees. Within a polygon it thence speaks of congruent vertex angles.None the less its usage is even more specific. It applies to any polytope's vertices and there moreover asks that those are to be considered transient under the action of a symmetry.
Isogonalpolygons are therefore quite easily recognized as the ones of the sorta-P-b, where in thisCoxeter-Dynkin diagrama andb denote the alternating side lengths andP = π/ϕ withϕ being the centri-angle underneath a pair of consecutive sidesa andb. For closing reasons here obviouslyP needs to be a rational numberP = n/d, wheren represents the number of sides of typea (or alike ofb) andd is the winding number, i.e. counts how often the sequence of sides runs around the center point. Polygons with winding numberd>1 are also known as polygrams instead. A polygram of winding numberd is also called ad-stropic polygram.
Many of the most often used qualifiers for polytopes use isogonality for one of their restrictions. So all these classes of polytopes readily are isogonal ones.Just to mention some:regular polytopes (as well as vertex-regularcompounds),uniform polytopes,scaliform polytopes,noble polytopes, etc. –Obviously all vertices of an isogonal polytope live on a single (hyper)sphere. Esp. all isogonal polytopes have an individual, well-defined circumradius.
It should be emphasized however that vertex congruence and vertex transitive is not the same. Only the latter is what isogonal means. Simply consider theMiller's solid, which looks nearly like therhombicuboctahedron.In fact all vertex figures show up 1 triangle and 3 squares. But the former has only 4-fold axial symmetry and so its vertices fall into 2 orbits.While the other has higher symmetry which then allows to map any vertex figure onto every other.
In what follows we will now restrict toconvex isogonal polytopes generally. As no restriction was set onto the edge sizes, it becomes obvious that thekaleidoscopical construction of Wythoff can easily generalized to arbitrary edge lengths. For instance consider
a3b3ca3b4ca3b5c
where each edge sizea,b, andc can be varied independenty, even down to zero, which then makes several vertices coincide.Still all these cases remain isogonal. –Together with the according prismatic case this encompasses all fully mirror symmetrical cases.
But there are subsymmetrical cases too, where full mirror symmetry breakes down.Just to provide some examples below: eg.trapezoprisms with rotational glide-reflection (e.g.rectangular,ditriangular, ...)or pyritohedrallygyrated polyhedra (eg.pyritosnub tetrahedron,pyritosnub cube, ...).
Already these two depicted examples show that facets of isogonal polytopes need not be isogonal themselves.
Despite the infinitude of shapes of convex isogonal polyhedra (as there are eg. the above mentioned continuous deformations of edge ratios), there seem to existonly topological variations ofuniform polyhedra. This however becomes quite different in higher dimensions, where combinatorically new polytopes will occure, i.e. with no mere edge resizement relation to any of the uniforms.(And this statement surely isn't true for non-convex polyhedra either.)
Alternated faceting, also known assnubbing, can be considered to have two consecutive processes. The first is the mere alternated faceting (in its stricter sense) from any Wythoffian starting figure, the second is the resizement back to all unit edges, if at all possible. If the latter succeeds, then the result will be uniform again; but not all cases do. Those remainders would still be isogonal polytopes.Because of the number of independent edge types involved and the degrees of freedom of the respective space grows on different paces,beyond 3D most snubs would remain isogonal only. Purely isogonal snubs are:
s3s3s3s (snip / snad) | o3o3o4s (hex: → uniform)x3o3o4s (rit: → uniform)o3x3o4s (thex: → uniform)o3o3x4s (rit: → uniform)x3x3o4s (tah: → uniform)x3o3x4s (rico: → uniform)o3x3x4s (tah: → uniform)x3x3x4s (tico: → uniform)s3s3s4o (sadi: → uniform)s3s3s4x (pysnet)s3s3s4s (snet) | s3s3s5s (snixhi) | s3s4o3o (sadi: → uniform)s3s4o3x (prissi: → scaliform)s3s4x3o (srico: → uniform)s3s4x3x (prico: → uniform)s3s4s3s (sni(-co) / snoc) | s3s3s *b3s (sadi → uniform) |
s2s3s3s (snittap) | s2o3o4s (hex: → uniform)s2x3o4s (tuta: → scaliform)s2o3x4s (cope: → uniform)s2x3x4s (tope: → uniform)s2s3s4o (pikap / snittap)s2s3s4x (pysna)s2s3s4s (sniccap) | s2s3s5s (sniddap) | sns2sms (n,m-dap)sns2s2mx (n,2m-pap)s3s2s3s (triddap)s5s2s5/3s (gudap: → uniform) | s2s2sns (2,n-dap)s2s2s2nx (2,2n-pap)s2nx2xmo (= ynx xmo)s2s2s2s (hex: → uniform)s2s2s3s (ditdap)s2s2s4x (dispap)s2x2s4xs2s2s6x (dihipap)s4x2x3o (= w x x3o) |
Truncation cannot even be generally applied to uniform polytopes. For, although all vertices are alike (isogonal),not all (incident) edges would be necessarily of the same type and so the ratios of respective truncation depths are not well-defined.Asrectification only is meant to be the instance, where truncations happen to meet, the same problem then applies here too.(More details on that is being found already in the respective linked pages.)For thequasiregular polytopes however we generally have
trunc( ...-o-N-x-M-o-...) = ...-x(N,2)-N-y-M-x(M,2)-... rect( ...-o-N-x-M-o-...) = ...-x(N,2)-N-o-M-x(M,2)-...
i.e. this results at least within variants ofWythoffians.–But there might be uniform cases even beyond these quasiregular polytopes, where an additional outer symmetry mightunite the various edge types into a single class. Then for sure truncation and rectification should apply. Again the results then are usually isogonal only.
trunc(o3x3x3o ) =tadeca rect(o3x3x3o ) =redeca | trunc(o3x4x3o ) =ticont rect(o3x4x3o ) =recont |
trunc(x3o3o3x ) =tispid rect(x3o3o3x ) =respid | trunc(x3o4o3x ) =tispic rect(x3o4o3x ) =respic |
trunc(xNo oNx ) rect(xNo oNx )N=3 →tatriddip andretdipN=4 →tat andrit (both uniform)... | |
(Note thatx3o3x *b3x=o3x4o3o and thence the there possible outer symmetry already was covered by the remark on quasiregulars.The same applies forx4o o4x=o3o3o4x.))
Expansion – not in theStott reading of the term (then resulting in virtually allWythoffian polytopes),but rather in the Conway reading as the operation e (below mentioned asexp however), which expands say a regular polytopexPoQoRoby its dualoPoQoRx, then resulting inxPoQoRx – could well be considered a bit beyond that restricted setup too, even if it still shall result in an isogonal outcome.The restriction here would be that the starting polytope (to be expanded) has to be anoble polytope and that (the single type of) itsfacets should be isogonal themselve. Then the expanded polytope clearly happens to remain isogonal.And this then clearly would be independent of the actual expansion rate, as isogonality quite genaral does not account on being used edge sizes.
exp(o3x3x3o ) =sobcated | exp(o3x4x3o ) =sobcotic |
exp(o3x x3o ) =triddep | exp(oNx xNo ) =n-dep |
Thetegum sum is the hull of thecompound of its addends.Applying this to Wythoffian polytopes with centrally symmetric symmetry group diagram, adding on the according outer symmetry, obviously results in at least isogonal hulls.
In the following varieties the size of the lacing edge throughout clearly is determined already by the all over sizes of the componentedge sizes, i.e. by the requirement to result in a degenerate (zero height)segmentotope (fromwhich then only the hull elements will contribute). Thus that one does represent a dependent variable only.
ao3oo3oo3oa&#zb -bideca, b:a = sqrt(3/5) = 0.774597oo3ao3oa3oo&#zb -apid, b:a = sqrt(2/5) = 0.632456ao3bo3ob3oa&#zc -bited, c:a = sqrt[(2y2+2y+3)/5], where y = b:aao3ob3bo3oa&#zc - five topologies: all c:a = sqrt[(2y2-2y+3)/5], where y = b:a (0 < b:a < 1/2) -sabred (b:a = 1/2) -respid, c:a = 1/sqrt(2) = 0.707107, c:b = sqrt(2) = 1.414214 (1/2 < b:a < 3) -mabred (b:a = 3) -redeca, c:a = sqrt(3) = 1.732051, c:b = 1/sqrt(3) = 0.577350 (3 < b:a < ∞) -gabredab3oo3oo3ba&#zc -biped, c:|a-b| = sqrt(3/5) = 0.774597oo3ab3ba3oo&#zc -bimted, c:|a-b| = sqrt(2/5) = 0.632456ao3bc3cb3oa&#zd (a,b,c > 0, c < sqrt[a2+ab+b2]) -sobcated, d = sqrt[(3a2+2a(b-c)+2(b-c)2)/5]ab3co3oc3ba&#zd - seven topologies: all d:|a-b| = sqrt[(2y2+2y+3)/5], where y = c:|a-b| (a > b) -(1,4)-expanded bited (a = b) - (1,4)-expanded bamid (0 < c:(b-a) < 1/2) - (1,4)-expanded sabred (c:(b-a) = 1/2) - (1,4)-expanded respid (1/2 < c:(b-a) < 3) - (1,4)-expanded mabred (c:(b-a) = 3) - (1,4)-expanded redeca (3 < c:(b-a) < ∞) - (1,4)-expanded gabredab3cd3dc3ba&#ze |
ao3oo4oo3oa&#zb -bicont, b:a = sqrt[2-sqrt(2)] = 0.765367oo3ao4oa3oo&#zb -bamic, b:a = 2-sqrt(2) = 0.585786ao3bo4ob3oa&#zc -bitec, c:a = sqrt[(6-4 sqrt(2))y2+(6-4 sqrt(2))y+(2-sqrt(2)], where y = b:aao3ob4bo3oa&#zc - five topologies: all c:a = sqrt[(6-4 sqrt(2))y2-(6-4 sqrt(2))y+(2-sqrt(2)], where y = b:a (0 < b:a < 1/2) -sabric (b:a = 1/2) -respic, c:a = 1/sqrt(2) = 0.707107, c:b = sqrt(2) = 1.414214 (1/2 < b:a < 2+sqrt(2)) -mabric (b:a = 2+sqrt(2)) -recont, c:a = sqrt[2+sqrt(2)] = 1.847759, c:b = sqrt[(2-sqrt(2))/2] = 0.541196 (2+sqrt(2) < b:a < ∞) -gabricab3oo4oo3ba&#zc -bipec, c:|a-b| = sqrt[2-sqrt(2)] = 0.765367oo3ab4ba3oo&#zc -bimtec, c:|a-b| = 2-sqrt(2) = 0.585786ao3bc4cb3oa&#zd (a,b,c > 0, c < sqrt[a2+ab+b2]) -sobcotic, d = sqrt[a2(2-sqrt(2))+2a(b-c)(3-2sqrt(2))+2(b-c)2(3-2sqrt(2))]ab3co4oc3ba&#zdab3cd4dc3ba&#ze |
aoo3ooo3oao *b3ooa&#zc -ico → uniform, c:a = 1/sqrt(2) = 0.707107aoo3ccc3oao *b3ooa&#zd -spic → uniform, c:a = d:a = 1/sqrt(2) = 0.707107aco3ooo3oac *b3coa&#zd - two topologies: all d:a = sqrt[(y2-y+1)/2], where y = c:a (0 < c:a < 1 or 1 < c:a < ∞) -... (a = c) -rico → uniform, d:a (or d:c) = 1/sqrt(2) = 0.707107aco3ddd3oac *b3coa&#ze - two topologies with d > 0: all e:a = sqrt[(y2-y+1)/2], where y = c:a (0 < c:a < 1 or 1 < c:a < ∞) -... esp. (c:a = (1+sqrt(5))/2, d:a = 1):prissi, e:a = 1 (a = c) - acd3ooo3dac *b3cda&#zeacd3eee3dac *b3cda&#zf |
ao3oo oa3oo&#zb -triddit, b:a = sqrt(2/3) = 0.816497ao3bo oa3ob&#zcao3ob oa3bo&#zc - three topologies: all c:a = sqrt[2(y2-y+1)/3], where y = b:a (0 < b:a < 1/2 or 2 < b:a < ∞) -... (b:a = 1/2 or b:a = 2) -retdip, c:a (or c:b) = 1/sqrt(2) = 0.707107 (1/2 < b:a < 2) -... esp. (b:a = 1):triddap, c:a = c:b = sqrt(2/3) = 0.816497ab3oo ba3oo&#zc -triddet, c = |b-a| sqrt(2/3)ao3bc oa3cb&#zd (c = b+2a) -tatriddip, d:a = sqrt[(2y2+5y+5)/3], where y = b:a (c = b) -triddep, d:a = sqrt(2/3), independent of y = b:aab3cd ba3dc&#ze |
Some rare isogonals might occur assubsymmetrical diminishings of other uniform polytopes. This esp. occurs whenever the undiminished form happens to be the convex hull of acompound. Then it might happen that fewer components still provide an isogonal hull. Or that a diminishing at the vertices of one (or more) components
The lacing squares of ann-gonal duoprism form a polygonal approximation to a torus. Thence this surface can be cut open and streched out to form an n × n grid of squares. Here it becomes obvious that a line of integral inclination, when starting at some vertex and being modwrapped according to thecyclical identifications (corresponding to the former cuts), would run from vertex to vertex: one step to the right and d steps up (mod n).When remapping this pattern back onto then-n-dip, the hull of those points defines an isogonal polytope, then-d-stepprism, provided that 1 < d < n-1. (In the remaining cases those points become corealmic and the polytope would degenerate.)
In general the cells of those stepprisms are simplices of lowest possible symmetry. For example the7-2-stepprism.None the less, in rare cases it happens that higher symmetrical cells occure. E.g. the 5-2-stepprism is nothing but a sqrt(x2+f2) = sqrt[(5+sqrt(5))/2] = 1.902113 scaledpen,i.e. vertex inscribed into apedip.Or the 8-3-stepprism is nothing but a sqrt(x2+w2) = sqrt[4+2 sqrt(2)] = 2.613126 scaledhex,i.e. vertex inscribed into aodip.And if n and d would not be coprime, then there even might occure higher antiprismatic cells too.Btw. the6-2-stepprism happens to be well-known for other reasons: it is nothing but the dual oftriddip.In fact the same relation applies quite generally for the following sub-class: the (2n)-(n-1)-stepprism is just the n-duotegum, i.e. the dual of then-duoprism.Also the10-3-stepprism is well-known, as it is nothing but the dual ofdeca.
Besides the isogonal stepprisms themselves some further constructions could be done with those. First of all there is their according n-foldcompound within then-n-dip army.Or, if chiral, an according achiral compound of 2n components.But sometimes it happens that fewer than n components would still provide an isogonal result too.In these cases also the according hull, then resulting in an accordingly diminishedn-n-dip,produces aconvex isogonal polytope again. An example here is thepentadiminished pentagonal duoprism.
More genereally, the same constructions also could be applied ton-n-n-tips within 6D,giving rise ton-d1-d2-stepprisms or ton-n-n-n-quips within 8D,giving rise ton-d1-d2-d3-stepprisms (etc.) too.
In a very loose sense allpolychora with a swirl-symmetry could be considered swirlchora. A swirl-symmetry isgenerated by Clifford rotations, which as such are dissecting 4D into 2 orthogonal subspaces and applying a normal 2D rotation each within either subspace at the same time, possibly at different paces.Esp. all theduoprisms would count in here.
In a much more specific sense however this term is used in close relation to the Hopf fibration: That one assures that there is a bijection between the pointson a the sphere (2-sphere within 3D) and the great circles of the glome (3-sphere within 4D). Apolytwister then is the according mapped result of anorbiform polyhedron. In fact its vertices get mapped to according great circles, its edges into twisted (i.e. non-flat but smoothly curved) faces (then looking like a Möbius strip, but with full turn, thus still orientable), and the faces get mapped into so called twisters, which are solid rings bounded by those twisted faces and having thereby throughout the polygonal cross-section of the pre-image. –Obviously we are way beyond polytopes here. Nonetheless these polytwisters still are kind of inbetween the fullyround shapes and elementalhierarchy of polytopes. Swirlchora now come in here as discretisations of those great circles back into polygons again and thus breaking up those partly smooth polytwisters into fully valid polytopal approximations.
While the most direct way of such a breaking up of the individual twisters would be to cut them orthogonally to a (ring-)inscribed polygon, but there clearly would be other ways too.E.g. when starting with aregular polyhedron and applying that described most direct way by means of a likewise regular polygon, the resultingpolyhedral sections of each twister would be globally identical chiralantiprisms. Thence such a figure will be isochoric (having identicalpolyhedral facets).
In fact, the connectedness of the mutually swirling individual twisters does furtherrestrict thatn after all. This thus brings back into play the formervertex figure of the starting polyhedron – in addition to the so far only considered faces thereof (the cross-sectionsof the twisters, i.e. the bases of the antiprisms). Because there also is a full inversion symmetry of the outcome of that fibration, we thus finally have to considedern = LCM(p, q, 2) for a starting polyhedron{p, q}.
The ones bowing to this restriction happen to be isogonal in addition and thence quallify to benoble at least.
The following table provides a listing of noblen-fold dividedpolyhedron-based swirlchora. Each is given in its isochoric description and mentioning its polygonal dissectioning.
| tetrahedral 4 great circles 4 twisters | octahedral 6 great circles 8 twisters | cubic 8 great circles 6 twisters | icosahedral 12 great circles 20 twisters | dodecahedral 20 great circles 12 twisters | great dodecahedral 12 great circles 12 twisters |
ico (n=6, regular) | trap-96 (n=12) | squap-72 (n=12) | trap-600 (n=30) | pap-360 (n=30) | sisp (n=10, uniform) |
J. Bowers defines in hisglossary the powertopes BP. However then the polytope P in the power is restricted to have brick symmetry.And furthermore the polytope B in the base has to have a center at least, bettermore if it is centro-symmetric.Then he outlines esp. the 4D cases, where B is some {p} and P = {8} (in its axes-aligned orientation).
As it happens all those latter ones can uniformly described in a closed form as ategum sum too:in fact they are nothing but
xw-p-oo wx-p-oo&#zywhere:x = 1, w = 1+sqrt(2) = 2.414214, y = 1/sin(π/p)
Indeed, the process requires to use thep-fold ring of the (long)prismsx.-p-o. w. .. from the first layer plus the orthogonalp-fold ring of (long)prisms.w .. .x-p-.o from the other layer,and then asks to connect them directly, i.e. in a lacing sense. These lacing elements thus arerectangular trapezoprisms,which here get described by the elementsxw .. wx ..&#zy, where the rectangular bases have edge sizesx andw each, while the lacing edge size will bey.
Applying the same techniques as above to honeycombs would yield here according isogonal euclidean space "polytopes" as well.
s4o3o4o (octet: → uniform) | s4x3o4o (rich: → uniform) | s4o3x4o (tatoh: → uniform) | s4o3o4s (octet: → uniform)s4o3o4x (sratoh: → uniform) |
o4s3s4o (bisch)= s3s3s3s3*a | s4s3s4o (serch)s4x3x4o (batch: → uniform)x4s3s4o (casch) | s4x3o4s (rusch)s4x3o4x (srich: → uniform)x4x3o4s (gratoh: → uniform) | s4s3s4s (snich)s4s3s4x (esch)s4x3x4x (grich: → uniform)s4x3x4s (gabreth)x4s3s4x (cabisch) |
s∞o2o3o6s (ditoh: → CRF) | s∞o2x3o6s (gyrich: → scaliform)s∞o2o3x6s (...: → limit-CRF) | s∞o2s3s6o (ditoh: → CRF) | s∞o2s3s6s (...)s∞o2s3s6x (...)s∞o2x3x6s (...: → limit-CRF) |
s∞x2o3o6s (editoh: → CRF) | s∞x2x3o6s (gyerich: → CRF)s∞x2o3x6s (...: → limit-CRF) | s∞x2s3s6o (editoh: → CRF) | s∞x2s3s6s (...)s∞x2s3s6xs∞x2x3x6s |
s∞o2s4o4o (octet: → uniform)s∞o2o4s4o (octet: → uniform) | s∞o2s4o4ss∞o2s4o4x (pacratoh: → CRF) | s∞o2s4s4o (...)s∞o2s4x4o (...: → limit-CRF)s∞o2x4s4o (...) | s∞o2s4s4s (...)s∞o2s4s4x (...)s∞o2s4x4s (...)s∞o2s4x4x (...: → limit-CRF)s∞o2x4s4x (...: → limit-CRF) |
s∞x2s4o4o (pextoh: → CRF)s∞x2o4s4o (pextoh: → CRF) | s∞x2s4o4ss∞x2s4o4x | s∞x2s4s4os∞x2s4x4os∞x2x4s4o | s∞x2s4s4ss∞x2s4s4xs∞x2s4x4ss∞x2s4x4xs∞x2x4s4x |
Note, since all 2D (non-holosnub)alternated facetings already resulted inuniform tilings, it becomes clear that according merehoneycomb productswithx∞o (orx∞x)generally result inuniform honeycombs too. Therefore those according infinite prismatical honeycombs have been omitted in the above listingsa priori. Rather only those have been included here, where some alternation in the axial direction takes place as well.
trunc(o4x3x4o ) =tabatch rect(o4x3x4o ) =rebatch | trunc(x4o3o4x ) =tich (uniform) rect(x4o3o4x ) =rich (uniform) | trunc(x3x3o3o3*a ) =tacytatoh rect(x3x3o3o3*a ) =recytatoh |
ao4oo3oo4oa&#zb -bichon, b:a = sqrt(3)/2 = 0.866025oo4ao3oa4oo&#zb -bamich, b:a = 1/sqrt(2) = 0.707107ao4bo3ob4oa&#zc -bitach, c:a = sqrt[2y2+2y sqrt(2)+3]/2, where y = b:aao4ob3bo4oa&#zc - five topologies: c:a = sqrt[2y2-2y sqrt(2)+3]/2, where y = b:a (0 < b:a < 1/sqrt(2)) -sabirch (b:a = 1/sqrt(2)) -rich (uniform: 'a' pseudo) (1/sqrt(2) < b:a < 3/sqrt(2)) -mabirch (b:a = 3/sqrt(2)) -rebatch (3/sqrt(2) < b:a < ∞) -gabirchab4oo3oo4ba&#zc -bipach, c:|a-b| = sqrt(3)/2 = 0.866025oo4ab3ba4oo&#zc -bimtich, c:|a-b| = 1/sqrt(2) = 0.707107ao4bc3cb4oa&#zdab4co3oc4ba&#zdab4cd3dc4ba&#ze |
ao3oo3oo3oa3*a&#zb -bithon, b:a = sqrt(3/8) = 0.612372ao3oo3oa3oo3*a&#zb -chon (regular: 'a' pseudo), b:a = 1/sqrt(2) = 0.707107ao3ob3bo3oa3*a&#zc - three topologies: c:a = sqrt[(3y2-2y+3)/8], where y = b:a (1 ≤ b:a < 3) -sobath esp. (b:a = 1):bamich a = b = q, c = x (b:a = 3) -rebtatoh (3 < b:a < ∞) -gobathao3bo3ob3oa3*a&#zc -..., c:a = sqrt[(3y2+2y+3)/8], where y = b:aao3bo3oa3ob3*a&#zc -..., c:a = sqrt[(1+y2)/2], where y = b:aab3oo3oo3ba3*a&#zc -bimteth, c:|a-b| = sqrt(3/8) = 0.612372... |
Applying the same techniques as above within 5D would yield here according isogonal polytera as well.
s3s3s3s3s (snod / snix) | s3s3s3s4o (snippit / snahin)x3x3x3x4s (gippit: → uniform)s3s3s3s4x (pysnan)s3s3s3s4s (snan) | s3s3s *b3s3s (snahin) | s2s3s3s4x (pysneta)s2s3s4o3o (sadiap)s2s3s4o3x (prissia) | s2no2o3o4s (...)s2nx2o3o4s (...)s4x2o3o4s (...)s4x2x3o4s (...)s6o2o3o4s (...)s6x2o3o4s (...) |
Just as for4D isogonal truncates within 5D too there are exceptional cases wheretruncationandrectification applies, even beyond the quasiregular polytera as pre-images.
trunc(x3o3o3o3x ) =tiscad rect(x3o3o3o3x ) =rescad | trunc(o3x3o3x3o ) = tasibrid rect(o3x3o3x3o ) =resibrid |
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