| Family | Form | Example | Family | Form | Example |
|---|---|---|---|---|---|
| rectangle 2-orthotope |
{ }×{ }  [2] | 2-cube t{2} |
rhombus 2-fusil |
{ }+{ }  [2] | 2-orthoplex |
| Polyhedra | Polychora | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Family | Form | Example | Family | Form | Family | Form | Example | Family | Form | Example |
| pyramid | {n}∨() [n,1] | triangular pyramid {3}∨() as tetrahedron |
digonal disphenoid |
{ }∨{ } [2] |
pyramid | {p,q}∨() [p,q,1] | tetrahedral pyramid {3,3}∨() as 5-cell |
{p}∨{ } [p,2,1] | tetrahedral pyramid {3}∨{ } as 5-cell | |
| Polyhedra | Polychora | |||||||
|---|---|---|---|---|---|---|---|---|
| Family | Form | Example | Family | Form | Example | Family | Form | Example |
| prism | { }×{n} =t{2,n}   [2,n] | square prism { }×{4} as cube  = t{2,4} |
prism | { }×{p,q} =t{2,p,q}   [2,p,q] | cubic prism { }×{4,3} as tesseract  = t{2,4,3} |
duoprism | {p}×{q} =rr{p,2,q} [p,2,q] | 4-4 duoprism {4}×{4} as tesseract = rr{4,2,4} |
| bipyramid (fusil) |
{ }+{n} [2,n] | square fusil { }+{4} as octahedron |
bipyramid (fusil) |
{ }+{p,q} [2,p,q] | octahedral fusil { }+{3,4} as 16-cell |
duopyramid (duofusil) |
{p}+{q} [p,2,q] | 4-4 duofusil {4}+{4} as 16-cell |
| antiprism | { }⨂{n} =h{2}s{n} =s{2,2n}  [2,n]+ [2+,2n] | triangular antiprism { }⨂{3} as octahedron = h{2}s{3} = s{2,6} |
antiprism | { }⨂{p,2q} =h{2}s{p,2q} [(2,p)+,2q] | snub octahedral antiprism { }⨂{3,4} = h{2}s{3,4} |
duoantiprism | {p}⨂{q} =s{p}s{q} =2s{2p,2,2q} [p,2,q]+ [2p,2+,2q] | 2-2 duoantiprism {2}⨂{2} as 16-cell = s{2}s{2} = 2s{4,2,4} |
| trapezohedron (antifusil) |
{ }⨁{n} [2,n]+ [2+,2n] | triangular antifusil { }⨁{3} as cube |
antifusil | { }⨁{p,2q} [(2,p)+,q] | snub octahedral antifusil { }⨁{3,4} as icosahedral antifusil |
duoantifusil | {p}⨁{q} [p,2,q]+ [2p,2+,2q] | 2-2 duoantifusil {2}⨁{2} as tesseract |
| Polyhedra | Polychora | |||
|---|---|---|---|---|
| Family | Example | Image | Family | Example |
| Prism { }×{n} [2,n], order 4n |
Square prism { }×{4} as cube [2,4], order 16 |  |
Duoprism {p}×{q} [p,2,q], order 4pq [[p,2,p]], order 8p2 |
4-4 duoprism {4}×{4} as tesseract [4,2,4], order 64 [[4,2,4]], order 128 |
| Fusil { }+{n} [2,n], order 4n |
Square fusil { }+{4} as octahedron [2,4], order 16 |  |
Duofusil {p}+{q} [p,2,q], order 4pq [[p,2,p]], order 8p2 |
4-4 duofusil {4}+{4} as 16-cell [4,2,4], order 64 [[4,2,4]], order 128 |
| Antiprism { }⨂{n} [2,n]+, order 2n [2+,2n], order 4n |
Triangular antiprism { }⨂{3} as octahedron [2,3]+, order 6 [2+,6], order 12 |  |
Duoantiprism {p}⨂{q} [p,2,q]+, order 2pq [2p,2+,2q], order 8pq [[2p,2+,2p]], order 16p2 |
2-2 duoantiprism {2}⨂{2} as 16-cell [2,2,2]+, order 8 [4,2+,4], order 32 [[4,2+,4]], order 64 |
| Antifusil { }⨁{n} [2,n]+, order 2n [2+,2n], order 4n |
Triangular antifusil { }⨁{3} as cube [2,3]+, order 6 [2+,6], order 12 |  |
Duoantifusil {p}⨁{q} [p,2,q]+, order 2pq [2p,2+,2q], order 8pq [[2p,2+,2p]], order 16p2 |
2-2 duoantifusil {2}⨁{2} as tesseract [2,2,2]+, order 8 [4,2+,4], order 32 [[4,2+,4]], order 64 |
Pyramid, fusil, prism
edit| Pyramid | Fusil | Prism | Type |
|---|---|---|---|
| Operator | Join | Rhombic sum | Rectangular product |
| Symbol | P ∨ Q | P + Q | P × Q |
| Dimension | a+b+1 | a+b | |
| 1D | ( ) ∨ ( ) = 2.( ) Ditel [ ] |
{ } Ditel [ ] | |
| 2D | ( ) ∨ { } Isosceles triangle [1] |
{ } + { } = 2{ } Rhombus [2] |
{ } × { } = { }2 Rectangle [2] |
| 3D | { } ∨ { } = 2.{ } Digonal disphenoid [1,2] |
{ } + {p} p-gonal fusil (p bipyramid) [2,p] |
{ } × {p} p-gonal prism (p prism) [2,p] |
| 4D | { } ∨ {p} p-gonal pyramid (p pyramid) [1,2,p] |
{p} + {q} p-gonal q-gonal fusil (p-q duofusil) [p,2,q] |
{p} × {q} p-gonal q-gonal prism (p-q duoprism) [p,2,q] |
| 2{p} p-gonal double fusil (p-p duofusil) [[p,2,q]] |
{p}2 p-gonal double prism (p-p duoprism) [[p,2,q]] | ||
| 5D | {p} ∨ {q} p-gonal q-gonal pyramid (p-q duopyramid) [1,p,2,q] | ||
| 2.{p} p-gonal double pyramid (p-p duopyramid) [1,[p,2,p]] | |||