Klitzing wrote:At least for the quadrangular pyramid the answer is easy:

you just have the restriction that the (planar) quadrangle has to have a circum circle.

Then any tip atop - what so ever - would define an according circum sphere.

--- rk

Hm, that opens some possibilities...

For uniform honeycombs, all four of the side edges must correspond to even polygons, but otherwise there seems to be a lot of options.

EDIT: Except that some pyramids might still be excluded because of impossible combination of side edges? Or is any combination of chords permissible?

The quadrangles that work for uniform tesselations are:

AAAA - a arbitrary - all four side edges must be the same. All four triangles are ASS.

AAAB - antiprismatic form - a divisible by 3 - a/a and a/b side edges could be same or different.

same: side triangles ASS, ASS, ASS, BSS

different: side triangles AST, ASS, AST, BTT - a must be even for this type to fit.

AAAB - hex form - a divisible by 6 - thinking about it, a hexagon in (6,6,6,3) has three different kinds of edges, so there are three possible side lengths here:

ASU, AST, ATU, BUU

Then we have three options with two side lengths:

AST, ASS, AST, BTT

ASS, AST, AST, BSS

AST, AST, ATT, BTT

And standard ASS, ASS, ASS, BSS with one side length.

Both these forms of AAAB have a special case with a=b, further options for AAAA when a is divisible by 6.

AABB - Three possible side lengths: a/a, a/b and b/b

AST, AST, BTU, BTU

With two-length options:

ASS, ASS, BST, BST

AST, AST, BST, BST

AST, AST, BTT, BTT (same as first option, just with A and B switched).

And one-length:

ASS, ASS, BSS, BSS

if a=b, additional options for AAAA with even a appear.

ABAB - only one type of edges, so only one possible side length: ASS, BSS, ASS, BSS

AABC - mixed form with a divisible by 4 and b and c even. a/a, a/b, b/c and a/c side edges could be all different. Leads to AAAB mixed type (for a = b or a = c), additional AABB options (for b = c) and additional AAAA options for (a = b = c).

ABAC - symmetrical, so only two possible side edges, a/b and a/c. Leads to AAAB semiregular type (for a = b or a = c), additional ABAB options (for b = c) and additional AAAA options for (a = b = c).

ABCD - all four types of edges can be different. Leads to AABC semiregular type (for two adjacent values equal), additional ABAC options (for two opposite values equal), additional AAAB semiregular, additional AABB, additional ABAB and additional AAAA.

So, all in all, we have:

AAAA - fully symmetrical with one side length, AAAB antiprismatic subtype with two, AAAB hex subtype with two, AABB subtype with up to three, AABC mixed subtype with up to four, ABAC subtype with two, ABCD subtype with up to four.

AAAB - semiregular - ABAC type with up to two side lengths, ABCD type with up to four.

AAAB - mixed - AABC type with up to four side lengths

AAAB - antiprismatic - fully symmetrical with up to two side lengths.

AAAB - hex - fully symmetrical with up to two side lengths.

AABB - fully symmetrical with up to three side lengths, AABC mixed type with up to four, ABCD type with up to four.

ABAB - fully symmetrical with one side length

AABC - semiregular - ABCD type with up to four side lengths

AABC - mixed - fully symmetrical with up to four side lengths

ABAC - fully symmetrical with up to two side lengths, ABCD type with up to four.

ABCD - fully symmetrical with up to four side lengths