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
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  
general polytopal classes:
Johnson solids  
uniform relative:
oct  
variations:
xxo&#q   ox4oo&#q  
general polytopal classes:
Johnson solids   segmentohedra   fundamental lace prisms  
External
links
wikipedia   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.

The rectangular pyramid version further is isomorphic to the bowtie pyramid (bobipyr).


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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