CRF polychora discovery project (Meta, 13)

From Hi.gher. Space

(Difference between revisions)

Latest revision as of 05:05, 9 March 2019

See also: List of convex regular-faced polyhedra

This page documents an ongoing project to discover as many CRF polychora as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.

Other than infinite series, the non-Johnson CRF polyhedra are the regular polyhedra and the Archimedean polyhedra. The Johnson solids can be roughly divided into four categories: prismatoids (includes pyramids, cupolae, rotundae, and elongated/bi/gyro forms of the above), augmented polyhedra, diminished polyhedra and crown jewels. We will use the same categorizations here.

Richard Klitzing enumerated a list of 177 segmentochora, which are the orbiform CRF polychora. However, these overlap with several different categories below, so the segmentochora will not be considered a category of their own. TODO: find out how many segmentochora are not yet counted below.

Discovery index (D numbers)

As of February 2014, new CRF discoveries are assigned a discovery index (aka D number), as a way of uniquely identifying the discovery without committing to a specific categorization or naming of it, which may not be feasible due to insufficient information at the time of discovery.

The discovery index page serves as the authoritative list of D number assignments.

Prismatoids

Total in this section (excluding stacks): 264

The first 64 CRF polychora are the convex uniform polychora, which can be divided up into 9 pyromorphs, 9 xylomorphs, 12 stauromorphs (not 15, because three were already covered as xylomorphs), 15 rhodomorphs, the snub demitesseract and the grand antiprism, and 17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph).
A further 92 CRF prisms are possible, based on each of the 92 Johnson solids. This brings the running total to 156. Some of these prisms are included in Klitzing's list of segmentotopes, but some are not due to their corresponding Johnson solids not being orbiform (having all vertices lie on a sphere).
There are 30 CRF monostratic cupolic forms (or just cupolae). These are constructed by placing uniform polyhedra from the same symmetry group (except in one special case, the snubdis antiprism) in each plane.
There are 10 bicupolic rings (not 12, because 2 are already counted).
There are 62 more prismatoid forms (not 64, because 2 are already counted).
There are some bistratic polychora formed by stacking monostratic polychora together, but these have not been counted up yet.
There are 6 ursachora.

Diminishings

Main article: CRFP4DP/Diminishings

Some diminishings of uniform polychora produce various segmentotopes and CRF polychora.
There are many BT polychora.
There are three known non-uniform convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
There are 18 diminishings of the xylochoron (not 19, because one is the tesseract).
The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
There are some modified bisected 600-cells, lunae (wedge-like multiply-bisected 600-cells) and rotundae.

Augmentations

Main article: CRFP4DP/Augmentations

The pyrochoron has a single augmentation, consisting of two pyrochora joined cell-to-cell.
The tesseract has 14 augmentations with cubical pyramids, one of which corresponds with the xylochoron. However, these augmentations are also a subset of the xylochoron's diminishings, so their count should not be included under the present category.
There are some modified tesseract augmentations and augmentations of truncated tesseracts.
There are probably many other augmentable uniform polychora, these have yet to be explored.
The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress. Preliminary calculations indicate that augmentation with n-gonal magnabicupolic rings number in the millions, due to combinatorial explosion.

Gyrations

wintersolstice discovered that some CRFs can be gyrated in various ways.

Modified Stott expansions

Arguably, the CRFs described below could be classified as crown jewels, even though since their initial discovery they have been found in large numbers, which somewhat defeats the label "crown jewels".

Partial Stott-expansions

Main article: Partial Stott-expansion

Klitzing discovered in 2013 that some infinite families can be expanded according to a lower symmetry group, giving new polytopes. Expanding a 16-cell by a 4-fold subsymmetry, for example, produces a CRF polychoron best described as the convex hull of a tesseract and an octagon, both centered on the origin. This represents the first step in a series of partial Stott expansions that eventually yields the runcinated tesseract (x4o3o3x).

EKF polytopes

In 2014 quickfur discovered a derivation of a bilbiro from an icosahedron. This led to the discovery of various partial expansions of the hydrochoron. This derivation is not merely a partial Stott expansion, but a modification of partial Stott expansion by adding an initial faceting step before the actual Stott expansion. Student91 discovered an underlying general scheme where a polytope (usually uniform or regular) can be partially faceted according to some subsymmetry, and then Stott-expanded according to the same symmetry in order to restore convexity, often resulting in novel CRFs. The faceting is done by "punning" a CD node label with an equivalent label that has a negative value, thereby producing a non-convex faceting, which is then restored to 0 by the subsequent Stott expansion.

In 3D, the icosahedron, for example, can be faceted according to a 2-fold subsymmetry and then expanded, producing a bilbiro (J91). The same process applied to a 3-fold subsymmetry produces the triangular hebesphenorotunda (J92). Applied to a 5-fold subsymmetry, this process produces J32, the pentagonal orthocupolarotunda.

This process has come to be known by the acronym EKF (expando-kaleido-faceting). When applied to various 4D uniform polytopes, primarily those in the 120-cell family, the EKF operation produces a large number of new CRFs, many of them containing J32, J91, and/or J92 cells. A good number, probably the majority, of crown jewels in the D4.x numbering scheme are EKF polytopes.

Infinite families

Main article: CRFP4DP/Infinite families

The obvious infinite family is that of the m,n-duoprisms (m ≥ n ≥ 3).
There is also an infinite family of prisms of the n-gonal antiprisms.
Mrrl discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two square antiprisms, one cube, four tetrahedra and four square pyramids. Details can be found in this post. In general, members of this family consists of two n-gonal antiprisms and an n-gonal prism, forming a 3-membered ring, with n tetrahedra and n square pyramids filling in the lateral gaps, for all n ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of biantiprismatic rings, and the specific term is the n-gonal biantiprismatic ring, e.g. square biantiprismatic ring. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the wedges.
The n-gonal pyramid antiprisms (n-gonal pyramid || inverted gyro n-gonal pyramid) are CRF, and for n=4 and n=5, non-orbiform. (For n=3, it is identical to the 16-cell.) They are identical to the n-antiprism bipyramid.

Pages in this category (7)

CRFP4DP/Stacks

@@ Line 1: / Line 1: @@
+<[#ontology [kind meta] [cats 4D CRF Polytope]]>
 {{selfref|See also: [[List of convex regular-faced polyhedra]]}}
 This page documents an ongoing project to discover as many [[Polytope#Convex regular-faced polytope|CRF polychora]] as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.
-== Summary ==
+Other than infinite series, the non-Johnson CRF polyhedra are the regular polyhedra and the Archimedean polyhedra. The Johnson solids can be roughly divided into four categories: prismatoids (includes pyramids, cupolae, rotundae, and elongated/bi/gyro forms of the above), augmented polyhedra, diminished polyhedra and [[crown jewel]]s. We will use the same categorizations here.
-{| class='wikitable'
-!Class||Expression||Value||Total
-|-
-|Convex uniform polychora||9 + 9 + 12 + 15 + 17 + 2||64||64
-|-
-|Prisms of Johnson solids||92||92||156
-|-
-|Prismatoid forms||13×4 - 2||50||206
-|-
-|Cupolae of regular polyhedra||3×7 + 2×4 - 1 + 6||34||240
-|-
-|Bicupolic rings||3×3||9||249
-|-
-|Rotundae||3×4||12||261
-|-
-|Duoprisms augmented with pyramids||See below||1633||1894
-|-
-|Convex Non-Uniform Scaliform||4||4||1898
-|}
+Richard Klitzing enumerated a list of 177 [[segmentochora]], which are the orbiform CRF polychora. However, these overlap with several different categories below, so the segmentochora will not be considered a category of their own. <span class='bad'>TODO: find out how many segmentochora are not yet counted below.</span>
+==Discovery index (D numbers)==
-== Convex uniform polychora ==
+As of February 2014, new CRF discoveries are assigned a [[discovery index]] (aka ''D number''), as a way of uniquely identifying the discovery without committing to a specific categorization or naming of it, which may not be feasible due to insufficient information at the time of discovery.
-The first 64 CRF polychora are the convex uniform polychora, which can be divided up into:
-*9 pyromorphs,
-*9 xylomorphs,
-*12 stauromorphs (not 15, because three were already covered as xylomorphs),
-*15 rhodomorphs,
-*17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph),
-*the [[snub demitesseract]] and the [[grand antiprism]].
-== Richard Klitzing's segmentotopes ==
+The [[discovery index]] page serves as the authoritative list of D number assignments.
-Dr. Richard Klitzing enumerated the full set of [[segmentochoron|segmentotopes]]: CRF polychora constructed by the convex hull of two lower-dimensional polytopes placed in parallel hyperplanes spaced appropriately so that the result will have equal-length edges.
+== Prismatoids ==
+{{selfref|Total in this section (excluding stacks): 264}}
+*The first 64 CRF polychora are the [[List of uniform polychora|convex uniform polychora]], which can be divided up into 9 pyromorphs, 9 xylomorphs, 12 stauromorphs (not 15, because three were already covered as xylomorphs), 15 rhodomorphs, the [[snub demitesseract]] and the [[grand antiprism]], and 17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph).
+*A further 92 CRF prisms are possible, based on each of the 92 [[Johnson solids]]. This brings the running total to 156. Some of these prisms are included in Klitzing's list of [[segmentotope]]s, but some are not due to their corresponding Johnson solids not being [[orbiform]] (having all vertices lie on a [[sphere]]).
+*There are 30 [[CRFP4DP/Monostratic cupolic forms|CRF monostratic cupolic forms]] (or just ''cupolae''). These are constructed by placing uniform polyhedra from the same symmetry group (except in one special case, the [[snubdis antiprism]]) in each plane.
+*There are 10 [[bicupolic ring]]s (not 12, because 2 are already counted).
+*There are 62 [[CRFP4DP/More prismatoids|more prismatoid forms]] (not 64, because 2 are already counted).
+*There are some [[CRFP4DP/Stacks|bistratic polychora formed by stacking monostratic polychora together]], but these have not been counted up yet.
+*There are 6 [[ursachora]].
-=== Prisms of Johnson solids ===
+== Diminishings ==
-There are 92 [[Johnson solids]]. Each one has a prism which is a CRF polychoron, bringing the running total to 156.
+{{selfref|Main article: [[CRFP4DP/Diminishings]]}}
+*Some diminishings of uniform polychora produce various segmentotopes and CRF polychora.
+*There are many [[BT polychora]].
+*There are three '''known''' non-uniform convex scaliform (equilateral and vertex transitive) polytopes. All their ridges are regular.
+*There are 18 diminishings of the xylochoron (not 19, because one is the tesseract).
+*The tesseract augmentations are precisely those that are only ortho or para; all meta-diminishings are not augmented tesseracts; there are 6 of these.
+*The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
+*There are some modified bisected 600-cells, lunae (wedge-like multiply-bisected 600-cells) and rotundae.
-=== Prismatoid forms ===
+== Augmentations ==
-We can generate 52 CRF polychora by all possible combinations of {[[tetrahedron]], [[cube]], [[octahedron]], [[icosahedron]], [[square antiprism]], [[pentagonal antiprism]], [[triangular prism]], [[pentagonal prism]], [[square pyramid]], [[pentagonal pyramid]], [[diminished icosahedron]], [[metabidiminished icosahedron]], [[tridiminished icosahedron]]} × {pyramid, bipyramid, elongated pyramid, elongated bipyramid}. However, two of these - the "tetrahedral pyramid" and the "octahedral bipyramid" - are already covered as the [[pyrochoron]] and the [[aerochoron]] respectively, leaving us with 50 new CRF polychora. This brings the running total to 206.
+{{selfref|Main article: [[CRFP4DP/Augmentations]]}}
+*The [[pyrochoron]] has a single augmentation, consisting of two pyrochora joined cell-to-cell.
+*The [[tesseract]] has 14 augmentations with cubical pyramids, one of which corresponds with the [[xylochoron]]. However, these augmentations are also a subset of the xylochoron's ''diminishings'', so their count should not be included under the present category.
+*There are some modified tesseract augmentations and augmentations of truncated tesseracts.
+*There are probably many other augmentable uniform polychora, these have yet to be explored.
+*The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
+*Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress. Preliminary calculations indicate that augmentation with n-gonal magnabicupolic rings number in the millions, due to combinatorial explosion.
-The remaining CRF polyhedra cannot generate pyramidal forms for one (or both) of the following reasons:
+== Gyrations ==
-*the polyhedron's vertices are further from its center than its edge length, thus any pyramid of it would require base-apex edge lengths longer than base-base edge lengths, and thus not be CRF;
+*wintersolstice discovered that [[CRFP4DP/Gyrations|some CRFs can be gyrated in various ways]].
-**note that this reason is implied if the polyhedron contains a [[contour]] with at least six edges, but the converse is not always true, e.g. in the case of the [[dodecahedron]]
-*the polyhedron cannot be [[inscribed in a sphere]], thus there is no point equidistant from all base points, thus any pyramid of it would have at least two different base-apex edge lengths, and thus not be CRF.
-[[wintersolstice]] originally proposed a list containing more polyhedra than those listed above, but this was incorrect due to the above reasons. He acknowledged that there was a mistake with the list some time ago, most likely realizing the same argument that has been written above, but did not give this explanation at the time.
+== Modified Stott expansions ==
-=== Cupolae of regular polyhedra ===
+Arguably, the CRFs described below could be classified as crown jewels, even though since their initial discovery they have been found in large numbers, which somewhat defeats the label "crown jewels".
-We can generate 21 CRF polychora from the possible combinations of {tetrahedral, cubic, dodecahedral} × {cupola, orthobicupola, gyrobicupola, elongated cupola, elongated orthobicupola, elongated gyrobicupola, antiprism}. There are an additional 8 forms constructed as {octahedral, icosahedral} × {cupola, orthobicupola, elongated cupola, elongated orthobicupola}, as these forms do not use both duals. This gives 29 shapes in total.
-The elongated cubic orthobicupola is the same as the runcinated tesseract, leaving us with 28 new CRF polychora.
+=== Partial Stott-expansions ===
+{{selfref|Main article: [[Partial Stott-expansion]]}}
+Klitzing discovered in 2013 that some infinite families can be expanded according to a lower symmetry group, giving new polytopes. Expanding a 16-cell by a 4-fold subsymmetry, for example, produces a CRF polychoron best described as the convex hull of a tesseract and an octagon, both centered on the origin. This represents the first step in a series of partial Stott expansions that eventually yields the runcinated tesseract (x4o3o3x).
-Each cupola is constructed as the spline from the base polyhedron to its [[extratruncate]]. In the case of gyrobicupolae, the "other end" of the polychoron is the dual of the base shape. In the case of antiprisms, the spline is directly from the base shape to its dual.
+===EKF polytopes===
+In 2014 quickfur discovered a derivation of a bilbiro from an icosahedron. This led to the discovery of various partial expansions of the hydrochoron. This derivation is not merely a partial Stott expansion, but a ''modification'' of partial Stott expansion by adding an initial faceting step before the actual Stott expansion.  Student91 discovered an underlying general scheme where a polytope (usually uniform or regular) can be partially faceted according to some subsymmetry, and then Stott-expanded according to the same symmetry in order to restore convexity, often resulting in novel CRFs.  The faceting is done by "punning" a CD node label with an equivalent label that has a negative value, thereby producing a non-convex faceting, which is then restored to 0 by the subsequent Stott expansion.
-<span style='color: red;'>The ability to construct these shapes with regular faces needs to be checked.</span>
+In 3D, the icosahedron, for example, can be faceted according to a 2-fold subsymmetry and then expanded, producing a [[bilbiro]] (J91). The same process applied to a 3-fold subsymmetry produces the triangular hebesphenorotunda (J92). Applied to a 5-fold subsymmetry, this process produces J32, the pentagonal orthocupolarotunda.
-Two copies of the icosahedron-dodecahedron antiprism can be fitted together by their dodecahedral bases; the relative sizes of the icosahedron and dodecahedron of equal edge length ensures that the result is convex, and therefore CRF. It consists of 2 icosahedral cells, 100 tetrahedra, and 24 pentagonal pyramids. An elongated form is obtained by inserting a dodecahedral prism. Both forms have augmented and biaugmented variants. The other polyhedron-dual antiprisms do not produce convex CRFs this way, so this is a unique combination giving 6 new CRF polychora in total.
+This process has come to be known by the acronym EKF (expando-kaleido-faceting).  When applied to various 4D uniform polytopes, primarily those in the 120-cell family, the EKF operation produces a large number of new CRFs, many of them containing J32, J91, and/or J92 cells. A good number, probably the majority, of crown jewels in the D4.x numbering scheme are EKF polytopes.
-=== Bicupolic rings ===
-Nine CRF polychora are available from the possible combinations of {triangle, square, pentagon} × {ortho, gyro, magna}. [[Keiji]] discovered the ortho- and gyro- forms, and [[quickfur]] discovered the magna- form. Keiji has dubbed these shapes [[bicupolic ring]]s in general, and the specific naming pattern is ''n''-gonal ''form''bicupolic ring, e.g. ''square orthobicupolic ring''.
-The ortho- and gyro- forms are constructed as in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16461 this post]. The magna- forms are constructed as in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16493 this post] (second-to-last paragraph).
-== Rotundae ==
-So far, three CRF rotundae have been discovered.
-Mrrl discovered that a CRF polychoron can be cut from the rectified 120-cell when diminishing the latter. This polychoron consists of 1 icosidodecahedron, 12 pentagonal rotundae, and 40 tetrahedra. It can be considered the 4D analogue of the 3D pentagonal rotunda.
-A similar CRF rotunda can be obtained from the cantellated 600-cell by a similar cutting, producing a polychoron with 1 icosidodecahedron, 12 pentagonal rotundae, 42 pentagonal prisms, 20 cuboctahedra, and 20 triangular cupolae.
-Mrrl also found that the top of the second rotunda can be diminished, to obtain another CRF rotunda with 1 truncated icosahedron, 12 pentagonal rotunda, 30 pentagonal prisms, and 40 triangular cupolae.
-These rotundae have birotunda forms as well as their corresponding elongates, giving 12 new CRF polychora in total.
-== Augmented uniform polychora ==
-Some of the uniform polychora can be ''augmented'' (have CRF pyramids erected on one or more of their cells) and still remain CRF. The criteria for such augmentations to be CRF are:
-* There must exist a CRF pyramid whose base is in the shape of the cell being augmented.
-* The sum of each dichoral angle of the pyramid's cells with its base and the dichoral angle of the cell being augmented and the corresponding neighbouring cell must be ≤ 180° in order to remain convex. If two adjacent cells are being augmented, then the sum of dichoral angles of two adjacent pyramid cells with their respective bases and the dichoral angle between the two augmented cells must be ≤ 180°.
-* In the case where the sum of dichoral angles is exactly 180°, adjacent cells in the augments will merge; in such a case, the merged cells must themselves be CRF.
-The [[pyrochoron]] has a single augmentation, consisting of two pyrochora joined cell-to-cell.
-The [[tesseract]] has 14 augmentations with cubical pyramids, one of which corresponds with the [[xylochoron]]. However, these augmentations are also a subset of the xylochoron's ''diminishings'', so their count should not be included under the present category.
-<s>The [[xylochoron]] has 20 augmentations ([http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16905#p16905 forum post]).</s> <span style="color:red">This has been found to be invalid, due to it being based on a faulty computation of the xylochoron's dichoral angle.</span>
-There are probably many other augmentable uniform polychora, these have yet to be explored.
-=== Augmented duoprisms ===
-==== With pyramids ====
-The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.
-The following lists the number of CRF polychora generated by augmenting duoprisms:
-<table style="margin-left:auto;margin-right:auto;">
-<tr><th>3,3-duoprism: </th><td>3</td></tr>
-<tr><th>3,4-duoprism: </th><td>5</td></tr>
-<tr><th>3,5-duoprism: </th><td>11</td></tr>
-<tr><th>3,6-duoprism: </th><td>4</td></tr>
-<tr><th>4,5-duoprism: </th><td>17</td></tr>
-<tr><th>4,6-duoprism: </th><td>4</td></tr>
-<tr><th>4,7-duoprism: </th><td>4</td></tr>
-<tr><th>4,8-duoprism: </th><td>7</td></tr>
-<tr><th>5,5-duoprism: </th><td>35</td></tr>
-<tr><th>5,6-duoprism: </th><td>12</td></tr>
-<tr><th>5,7-duoprism: </th><td>17</td></tr>
-<tr><th>5,8-duoprism: </th><td>29</td></tr>
-<tr><th>5,9-duoprism: </th><td>45</td></tr>
-<tr><th>5,10-duoprism: </th><td>77</td></tr>
-<tr><th>5,11-duoprism: </th><td>15</td></tr>
-<tr><th>5,12-duoprism: </th><td>25</td></tr>
-<tr><th>5,13-duoprism: </th><td>30</td></tr>
-<tr><th>5,14-duoprism: </th><td>48</td></tr>
-<tr><th>5,15-duoprism: </th><td>63</td></tr>
-<tr><th>5,16-duoprism: </th><td>98</td></tr>
-<tr><th>5,17-duoprism: </th><td>132</td></tr>
-<tr><th>5,18-duoprism: </th><td>208</td></tr>
-<tr><th>5,19-duoprism: </th><td>290</td></tr>
-<tr><th>5,20-duoprism: </th><td>454</td></tr>
-<tr><th>Total: </th><td>1633 augmentations</td></tr>
-</table>
-The 4,4-duoprism is omitted here, because it coincides with the tesseract, the augmentations of which are covered under another category.
-The sharp drop in the number of augmentations between the 3,5-duoprism and the 3,6-duoprism, between the 4,5-duoprism and the 4,6-duoprism, and between the 5,5-duoprism and the 5,6-duoprism is because pyramids of hexagonal (or higher) prisms cannot be CRF, since equilateral triangles tile the hexagon and so no hexagonal (or higher) pyramid can be formed without breaking the regular-faced requirement. Thus, only one of the duoprism's two rings can be augmented.
-The drop between the 5,10-duoprism and the 5,11-duoprism is caused by the fact that adjacent pentagonal prism pyramids erected on an n-membered duoprism ring are no longer convex after n=10, so from the 5,11-duoprism onwards only non-adjacent augmentations are permitted, thus reducing the number of possible combinations. Adjacent augments on the 5,10-duoprism have pentagonal pyramid cells that are coplanar, thus merging into a pentagonal bipyramid.
-Augments of the 5,20-duoprism have pentagonal pyramids coplanar with the adjacent pentagonal prism, so they merge into elongated pentagonal pyramids. If the next prism in the ring is also augmented, then another pentagonal pyramid is added to the coplanar cell, turning it into an elongated pentagonal bipyramid.
-No other duoprisms can be augmented with CRF pyramids and still remain convex.
-==== With other segmentochora ====
-Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress.
-So far, it has been found that some of the n,6-duoprisms, n,8-duoprisms, and n,10-duoprisms can be augmented by the segmentotopes m-gon||2m-prism for m=3,4,5. These augments induce an ''orientation'' on both duoprism rings, reducing symmetry and increasing the number of distinct augmentations. These augmentations have been enumerated for the following duoprisms:
-* 3,6-duoprism: The first ring can be augmented with triangle||hexagonal_prism and the second ring with triangular prism pyramids, albeit not at the same time. Total: 9 augmentations.
-* 4,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with cubical pyramid, albeit not at the same time. Total: 7 augmentations.
-* 5,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with pentagonal prism pyramid. First ring augments reduce the symmetries of the second ring by half, thus increasing the number of distinct combinations. Total: 64 augmentations.
-* 6,6-duoprism: First ring augmentable with triangle||hexagonal_prism; only one ring augmentable at a time. Total: 4 augmentations.
-* 3,8-duoprism: First ring augmentable with square||octagonal_prism; second ring cannot be augmented. Total: 5 augmentations.
-* 4,8-duoprism: First ring augmentable with square||octagonal prism; second ring with cubical pyramid (non-adjacent only). Both rings can be augmented simultaneously, but only when all augments on the first ring are in ''ortho'' orientation. Total: 44 augmentations.
-* 5,8-duoprism: TBD.
-The tesseract (considered as a 4,4-duoprism) can be augmented with 4 square pyramid prisms and 4 line||square segmentochora to form a CRF polychoron having 4 cubes, 16 triangular prisms, and 16 tetrahedra. This is somewhat different augmentation from what is considered above, in that the initial augmentation creates a non-convex polychoron, but the gaps can be filled in with CRF segmentochora to form a valid CRF polychoron. This shape can also be generated as the convex hull of a tesseract and an octagon. It was discovered by [[User:Quickfur|quickfur]] on 9 Jan 2012.
-== Diminished polychora ==
-Some regular polychora can be diminished to give CRF polychora. The 24-cell can be diminished into the tesseract by removing 8 square pyramids. Removing less than 8 pyramids in various configurations generates a number of distinct diminished 24-cells. It is also possible to remove square pyramids that do not correspond with facets of the tesseract, this generates a few more CRF polychora not included in the tesseract construction.
-The diminished 16-cell coincides with the octahedral pyramid (see Prismatoid forms above).
-The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).
 == Infinite families ==
-The obvious infinite family is that of the ''m'',''n''-duoprisms (''m'' ≥ ''n'' ≥ 3).
+{{selfref|Main article: [[CRFP4DP/Infinite families]]}}
+*The obvious infinite family is that of the ''m'',''n''-duoprisms (''m'' ≥ ''n'' ≥ 3).
-There is also an infinite family of prisms of the ''n''-gonal antiprisms.
+*There is also an infinite family of prisms of the ''n''-gonal antiprisms.
+*[[Mrrl]] discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two [[square antiprism]]s, one [[cube]], four [[tetrahedra]] and four [[square pyramid]]s. Details can be found in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16496 this post]. In general, members of this family consists of two ''n''-gonal antiprisms and an ''n''-gonal prism, forming a 3-membered ring, with ''n'' tetrahedra and ''n'' square pyramids filling in the lateral gaps, for all ''n'' ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of ''biantiprismatic rings'', and the specific term is the ''n''-gonal biantiprismatic ring, e.g. ''[[square biantiprismatic ring]]''. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes where they are numbered among the ''wedges''.
-[[Mrrl]] discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two [[square antiprism]]s, one [[cube]], four [[tetrahedra]] and four [[square pyramid]]s. Details can be found in [http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&p=16497#p16496 this post]. In general, members of this family consists of two ''n''-gonal antiprisms and an ''n''-gonal prism, forming a 3-membered ring, with ''n'' tetrahedra and ''n'' square pyramids filling in the lateral gaps, for all ''n'' ≥ 3. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of ''biantiprismatic rings'', and the specific term is the ''n''-gonal biantiprismatic ring, e.g. ''square biantiprismatic ring''. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes (they are known as ''wedges'' in Klitzing's terminology).
+*The n-gonal pyramid antiprisms (n-gonal pyramid || inverted gyro n-gonal pyramid) are CRF, and for n=4 and n=5, non-orbiform. (For n=3, it is identical to the 16-cell.) They are identical to the n-antiprism bipyramid.
-== Convex Non-Uniform Scaliform ==
-Outside the uniform polychora (uniformed-celled and vertex transitive) there are 4 '''known''' convex scaliform (equilateral and vertex transitive)
-Note: all the faces on a scaliform are regular
-they are:
-Bi-icositetradiminished 600-cell
-The 600-cell can be made as the convex hull of 5 24-cells removing the vertices of one and taking the convex hull, creates the the snub 24-cell, removing another (and taking the convex hull) creates this shape. It cells are 48 tridimished icosahedra
-Prismatorhombato snub 24-cell
-This is made by shrinking the 24 icosahedral cells of a snub 24-cell(all the tetrahedra are removed) Where icosahedra used to be joined by triangles they are now seperated by triangular prisms(96 in total), the holes in the shape are filled with 24 truncated tetrahedra and 96 triangular cupola.
-Truncated Tetrahedra Cupoliprism (one of Richard Klitzing segmentotopes)
-this is made by taking two Truncated Tetrahedra with hexagons lined up to triangles and then putting 8 triangular cuplolae on the hexagons and triangles, then filling in the holes with 5 tetrahedra.
-Swirlprismatodiminished Rectified 600-cell
-the Rectified 600-cell can be made as the convex hull of 6 600-cells, removed the vertices of one and take the convex hull and you get this shape. It's cells are 600 square pyramids, 120 pentagonal prisms and 120 pentagonal antiprisms.