Square (EntityTopic, 20)
From Hi.gher. Space
A square is a two-dimensional hypercube.
Squares are the most common base for two-dimensional manifolds and polyominoes.
The square also forms one of the three regular tilings of two-dimensional space, the square tiling.
Coordinates
The Cartesian cordinates for a square of side 2, centered at the origin, are:
(±1, ±1)
Equations
- Variables:
l ⇒ length of edges of the square
- The radial slices θ of a square are:
[:x,:y] ⇒ digon of length lsin(45° + (θ % 90°)√2
- The cross sections of a square are:
[!x,!y] ⇒ digon of length l
- The hypervolumes of a square are:
perimeter (total edge length) = 4l
area = l^{2}
Homology groups
Any unstated homology group is the trivial group 0.
- 0-frame (four points)
- H_{0} = 4ℤ
- 1-frame (four line segments)
- H_{0} = ℤ, H_{1} = ℤ
- 2-frame (solid square)
- H_{0} = ℤ
Diamond
A diamond is the dual of a square when orientation is preserved. In other words, it's a square rotated around the origin by 45 degrees. it can be considered to be represented as <II>.
The diamond should not be confused with the rhombus.
Brick
The diamond is a brick with unique points at (1,0) and (0,1). It represents the tegum product and the SUM function.
Brick
The square is a brick with one unique point at (1,1). It represents the Cartesian product and the MAX function.
Dissection
The square of side 2 may be dissected into 4× triangle with sides 2, √2, √2 and angles 45°, 45°, 90°.
Incidence matrix
Usage as facets
- prism: 6× 1-facets of a cube
- 12× 1-facets of a rhombic dodecahedron
- 30× 1-facets of a rhombic triacontahedron
- 6× 1-facets of a stauromesohedron
- 10× 1-facets of a decagonal prism
- 6× 1-facets of a hexagonal prism
- 8× 1-facets of a octagonal prism
- 5× 1-facets of a pentagonal prism
- 2× 1-facets of a square antiprism
- 8× 1-facets of a tetragonal trapezohedron
- 3× 1-facets of a triangular prism
- 6× 1-facets of a truncated octahedron
- 60× 1-facets of a deltoidal hexecontahedron
- 24× 1-facets of a deltoidal icositetrahedron
- 30× 1-facets of a rhodoperihedron
- pyramid: 1× 1-facets of a square pyramid (named lateral: faces)
- 6× 1-facets of a stauroperihedron
- 12× 1-facets of a stauroperihedron
- 30× 1-facets of a rhodopantohedron
- 6× 1-facets of a snub cube
- 12× 1-facets of a stauropantohedron
- 3× 1-facets of a associahedron
- 4× 1-facets of a digon-unpinched pentagonal prism
- 4× 1-facets of a parabiorthotriangulated cube
- 4× 1-facets of a parabiorthotriangulated cube_
- 3× 1-facets of a J3 dual
- 4× 1-facets of a J4 dual
- 5× 1-facets of a J5 dual
- 5× 1-facets of a pentagonal cupola
- 1× 1-facets of a rhombocupolawedge
- 2× 1-facets of a rhombopyrawedge
- 2× 1-facets of a rhombopyrawedge
- 1× 1-facets of a square cupola
- 4× 1-facets of a square cupola
- 3× 1-facets of a triangular cupola
- 8× 1-facets of a J37 (named slopes)
- 2× 1-facets of a J37 (named ends)
- 8× 1-facets of a J37 (named sides)
- 16× 1-facets of a J37 dual
- 8× 1-facets of a J37 dual
- 3× 1-facets of a J63 dual
- 2× 1-facets of a bilunabirotunda
- 2× 1-facets of a J91 dual
- 8× 1-facets of a J91 dual
- 2× 1-facets of a triangulated triangular prism
- 4× 1-facets of a hexagonal cupolawedge
- 2× 1-facets of a (dual of metabidiminished icosahedron)
- 4× 1-facets of a (dual of metabidiminished icosahedron)
- 1× 1-facets of a digon-unpinched square pyramid
- 2× 1-facets of a metabigyrotriangulated cube
- 2× 1-facets of a metabigyrotriangulated cube
- 1× 1-facets of a transposed diminished cube
- 2× 1-facets of a transposed diminished cube
- 6× 1-facets of a J92 dual
- 3× 1-facets of a J92 dual
- 6× 1-facets of a J92 dual
- 3× 1-facets of a J92 dual
- 3× 1-facets of a triangular hebesphenorotunda (named square)
- 2× 1-facets of a biunpinched cube
- 1× 1-facets of a biunpinched cube
- 2× 1-facets of a biunpinched cube
- 1× 1-facets of a pentagonal cupolawedge
- 1× 1-facets of a pentagonal cupolawedge
- 1× 1-facets of a pentagonal transcupolawedge
- 1× 1-facets of a pentagonal transcupolawedge
- 2× 1-facets of a square cupolawedge
- 2× 1-facets of a square cupolawedge
- 1× 1-facets of a square cupolawedge
- 2× 1-facets of a square cupolawedge
- 1× 1-facets of a square transcupolawedge (named roob)
- 2× 1-facets of a square transcupolawedge (named pgcb)
- 1× 1-facets of a triangulated pinched triangular cupola (named red)
- 1× 1-facets of a triangulated pinched triangular cupola (named orange)
- 24× 2-facets of a geochoron
- 9× 2-facets of a duotrianglinder
- 6× 2-facets of a (dual of octahedral prism)
- 12× 2-facets of a octahedral prism
- 6× 2-facets of a tetrahedral prism
- 72× 2-facets of a trixylodiminished hydrochoron
- 6× 2-facets of a cubic pyramid (named lateral: faces)
- 3× 2-facets of a triangular diprism (named joins 2 cubes)
- 12× 2-facets of a triangular diprism (named joins triangular prism to cube)
- 9× 2-facets of a (dual of triangular octagoltriate) (named square)
- 18× 2-facets of a (dual of triangular octagoltriate) (named trapezium)
- 18× 2-facets of a triangular octagoltriate (named rectangles in prisms)
- 18× 2-facets of a triangular octagoltriate (named trapeziums in antifrustoids)
- 1× 2-facets of a (dual of square pyramid prism)
- 2× 2-facets of a square pyramid prism
- 4× 2-facets of a square pyramid prism
- 4× 2-facets of a square pyramid prism
- 4× 2-facets of a gyrated octahedral prism (named around middle)
- 2× 2-facets of a gyrated octahedral prism (named in-out middle)
- 8× 2-facets of a gyrated octahedral prism (named walls)
- 2× 2-facets of a K4.13 dual
- 4× 2-facets of a K4.13 dual
- 8× 2-facets of a K4.13 dual
- 1× 2-facets of a K4.8
- 4× 2-facets of a K4.8
- 2× 2-facets of a K4.8 dual
- 4× 2-facets of a (dual of square biantiprismatic ring)
- 2× 2-facets of a square biantiprismatic ring (named end)
- 4× 2-facets of a square biantiprismatic ring (named side)
- 1× 2-facets of a square biantiprismatic ring (named mid)
- 60× 2-facets of a castellated rhodoperihedral prism
- 144× 2-facets of a (dual of truncated snub demitesseract)
- 48× 2-facets of a D4.11
- 96× 2-facets of a D4.11 dual
- 48× 2-facets of a D4.11 dual
- 8× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 2× 2-facets of a bilunabirotunda pseudopyramid (named cuboid end)
- 4× 2-facets of a bilunabirotunda pseudopyramid (named vert-to-roof)
- 60× 2-facets of a castellated rhodopantohedral prism (named 2 end)
- 120× 2-facets of a castellated rhodopantohedral prism (named 3 diag)
- 120× 2-facets of a castellated rhodopantohedral prism (named 5 diag)
- 60× 2-facets of a castellated rhodopantohedral prism (named 3-5 vert)
- 3× 2-facets of a D4.16 (named sweep of trip)
- 3× 2-facets of a D4.16 dual
- 6× 2-facets of a D4.16 dual
- 60× 2-facets of a D4.3.2 dual
- 120× 2-facets of a D4.3.2 dual
- 8× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 1× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 8× 2-facets of a bitrigonal diminished pyrocantichoron
- 2× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 8× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 4× 2-facets of a (4D analog of J37)
- 32× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 4× 2-facets of a (4D analog of J37)
- 4× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (dual of 4D analog of J37)
- 16× 2-facets of a (dual of 4D analog of J37)
- 16× 2-facets of a (dual of 4D analog of J37)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named square)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named tola-square to apex)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named square-bot to apex)
- 6× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oopp)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oogg)
- 6× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oooo)
- 12× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 6× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 6× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oooo)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oogg)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oopp - square)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named tola-square to apex)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named square-bot to apex)
- 24× 2-facets of a D4.7 (named oycb)
- 12× 2-facets of a D4.7 (named ggcc)
- 6× 2-facets of a D4.7 (named cccc)
- 12× 2-facets of a D4.7 dual
- 12× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 6× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 12× 2-facets of a D4.7 dual
- 12× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 12× 2-facets of a D4.7 dual
- 80× 3-facets of a geoteron
- 18× 3-facets of a triangular triprism (named joins base-cubes with 2 sweep-cubes)
- 36× 3-facets of a triangular triprism (named joins 1 triangular prism with 2 sweep-cubes)
See also
Hypercubes |
point • digon • square • cube • geochoron • geoteron • geopeton |
Notable Dishapes | |
Flat: | triangle • square • pentagon • hexagon • octagon • decagon |
Curved: | circle |
2. 2 Circle | 3. 11 Square | 4. 1^{1} Triangle |
List of tapertopes |
a. | b. | 1a. II Square | 1b. (II) Circle | 2a. III Cube | 2b. (III) Sphere |
List of toratopes |
1. I Digon | 2. [II] Square | 3. (II) Circle |
List of bracketopes |