Acronym squippy, 4pyr, J1
Name tetragonal pyramid,
half of oct
|,>,O device line prism pyramid = ||>
 
 © ©
Circumradius 1/sqrt(2) = 0.707107
Vertex figures [34] = x4o, [3,3,4] = oq&#x
Volume sqrt(2)/6 = 0.235702
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dual (topologically selfdual, in different orientation)
Dihedral angles
  • between {3} and {3}:   arccos(-1/3) = 109.471221°
  • between {3} and {4}:   arccos(1/sqrt(3)) = 54.735610°
Confer
general pyramids:
n/d-py  
uniform relative:
oct  
variations:
xxo&#q   ox4oo&#q  
general polytopal classes:
Johnson solids   segmentohedra   fundamental lace prisms   lace simplices  
External
links
wikipedia   polytopewiki   mathworld   quickfur

Measurements are given for the square pyramid only. Even this square can be viewed as a general tetragon, like a (special) rectangle, a (special) rhomb. It is just a question of applied symmetry (for instance by coloring elements). Alternatively, tetragonal pyramids, according to the general aplication of such non-square bases, by means of differently sized edge lengths or non-right angles, can be considered too. In those cases measurements would clearly depend on the special form of that tetragonal base.

Squippy is an edge-faceting of oct, in fact it uses triangles of it and an equatorial square.


Incidence matrix according to Dynkin symbol

square pyramid = ox4oo&#x   → height = 1/sqrt(2) = 0.707107
(pt || {4})

tip  o. o.    | 1 * | 4 0 | 4 0
base .o .o    | * 4 | 1 2 | 2 1
--------------+-----+-----+----
lace oo4oo&#x | 1 1 | 4 * | 2 0
base .x ..    | 0 2 | * 4 | 1 1
--------------+-----+-----+----
coat ox ..&#x | 1 2 | 2 1 | 4 *
base .x4.o    | 0 4 | 0 4 | * 1

rectangular pyramid = ox ox&#x   → height = 1/sqrt(2) = 0.707107
(pt || {4})

tip  o. o.    | 1 * | 4 0 0 | 2 2 0
base .o .o    | * 4 | 1 1 1 | 1 1 1
--------------+-----+-------+------
lace oo oo&#x | 1 1 | 4 * * | 1 1 0
base .x ..    | 0 2 | * 2 * | 1 0 1
base .. .x    | 0 2 | * * 2 | 0 1 1
--------------+-----+-------+------
coat ox ..&#x | 1 2 | 2 1 0 | 2 * *
coat .. ox&#x | 1 2 | 2 0 1 | * 2 *
base .x .x    | 0 4 | 0 2 2 | * * 1

trapezical pyramid = oxx&#x   → all heights = sqrt(3)/2 = 0.866025

o..    | 1 * * | 2 2 0 0 0 | 1 1 2 0
.o.    | * 2 * | 1 0 1 1 0 | 1 0 1 1
..o    | * * 2 | 0 1 0 1 1 | 0 1 1 1
-------+-------+-----------+--------
oo.&#x | 1 1 0 | 2 * * * * | 1 0 1 0
o.o&#x | 1 0 1 | * 2 * * * | 0 1 1 0
.x.    | 0 2 0 | * * 1 * * | 1 0 0 1
.oo&#x | 0 1 1 | * * * 2 * | 0 0 1 1
..x    | 0 0 2 | * * * * 1 | 0 1 0 1
-------+-------+-----------+--------
ox.&#x | 1 2 0 | 2 0 1 0 0 | 1 * * *
o.x&#x | 1 0 2 | 0 2 0 0 1 | * 1 * *
ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 2 *
.xx&#x | 0 2 2 | 0 0 1 2 1 | * * * 1

rhombical pyramid

tip  | 1 * * | 2 2 0 | 4 0
base | * 2 * | 1 0 2 | 2 1
base | * * 2 | 0 1 2 | 2 1
-----+-------+-------+----
lace | 1 1 0 | 2 * * | 2 0
lace | 1 0 1 | * 2 * | 2 0
base | 0 1 1 | * * 4 | 1 1
-----+-------+-------+----
coat | 1 1 1 | 1 1 1 | 4 *
base | 0 2 2 | 0 0 4 | * 1

triangular wedge   → height = sqrt(2/3) = 0.816497
(line || {3})

top  | 2 * * | 1 1 1 0 0 | 1 1 1 0
base | * 1 * | 0 2 0 2 0 | 1 0 2 1
base | * * 2 | 0 0 1 1 1 | 0 1 1 1
-----+-------+-----------+--------
top  | 2 0 0 | 1 * * * * | 1 1 0 0
lace | 1 1 0 | * 2 * * * | 1 0 1 0
lace | 1 0 1 | * * 2 * * | 0 1 1 0
base | 0 1 1 | * * * 2 * | 0 0 1 1
base | 0 0 2 | * * * * 1 | 0 1 0 1
-----+-------+-----------+--------
coat | 2 1 0 | 1 2 0 0 0 | 1 * * *
coat | 2 0 2 | 1 0 2 0 1 | * 1 * *
coat | 1 1 1 | 0 1 1 1 0 | * * 2 *
base | 0 1 2 | 0 0 0 2 1 | * * * 1

fully asymmetric

top  | 1 * * * * | 1 1 1 1 0 0 0 0 | 1 1 1 0 0
base | * 1 * * * | 1 0 0 0 1 1 0 0 | 1 1 0 0 1
base | * * 1 * * | 0 1 0 0 1 0 1 0 | 1 0 1 0 1
base | * * * 1 * | 0 0 1 0 0 0 1 1 | 0 0 1 1 1
base | * * * * 1 | 0 0 0 1 0 1 0 1 | 0 1 0 1 1
-----+-----------+-----------------+----------
lace | 1 1 0 0 0 | 1 * * * * * * * | 1 1 0 0 0
lace | 1 0 1 0 0 | * 1 * * * * * * | 1 0 1 0 0
lace | 1 0 0 1 0 | * * 1 * * * * * | 0 0 1 1 0
lace | 1 0 0 0 1 | * * * 1 * * * * | 0 1 0 1 0
base | 0 1 1 0 0 | * * * * 1 * * * | 1 0 0 0 1
base | 0 1 0 0 1 | * * * * * 1 * * | 0 1 0 0 1
base | 0 0 1 1 0 | * * * * * * 1 * | 0 0 1 0 1
base | 0 0 0 1 1 | * * * * * * * 1 | 0 0 0 1 1
-----+-----------+-----------------+----------
coat | 1 1 1 0 0 | 1 1 0 0 1 0 0 0 | 1 * * * *
coat | 1 1 0 0 1 | 1 0 0 1 0 1 0 0 | * 1 * * *
coat | 1 0 1 1 0 | 0 1 1 0 0 0 1 0 | * * 1 * *
coat | 1 0 0 1 1 | 0 0 1 1 0 0 0 1 | * * * 1 *
base | 0 1 1 1 1 | 0 0 0 0 1 1 1 1 | * * * * 1

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