Low-dimension

The low dimensions, i.e. up to 2 dimensions, are the simplest. They are all accessible here.

## 1 dimension

### 1D polytope

A 1-dimensional polytope is a pair of points:

The Schläfli symbol for this is {}. The Coxeter diagram is .

### 1D honeycomb

The 1-dimensional honeycomb is an apeirogon. This is a single line, with equally-spaced ‘vertices’:

## 2 dimensions

If an apeirogon was bent round in an axis perpendicular to the apeirogon, such that it comes back on itself, it then forms a polygon, the 2-dimensional polytope.

If a polygon has an even number of edges, and opposite edges are parallel, it is a zonogon. Examples are: all even-edged regular polygons, rectangles, parallelograms, etc.

### Regular polygons

A regular polygon can be represented by a number

p

=

p

n

p

d

{\displaystyle p~=~{\frac {p\_{n}}{p\_{d}}}}

, where

p

n

{\displaystyle p\_{n}}

 is the number of edges of the polygon, and

p

d

{\displaystyle p\_{d}}

 is the number of times the edge goes around the polygon. This is also the density of the polygon.

If we denote the edge length as

2
l

{\displaystyle 2l}

, then the circumradius,

r

c

{\displaystyle r\_{c}}

 is

cosec
⁡

(

π
p

)

{\displaystyle \operatorname {cosec} \left({\frac {\pi }{p}}\right)}

, and the inradius

r

i

{\displaystyle r\_{i}}

 is

cot
⁡

(

π
p

)

{\displaystyle \operatorname {cot} \left({\frac {\pi }{p}}\right)}

.

For information on specific polyhedra, see the individual pages:
1. Triangle ({3})

2. Square ({4})

3. Pentagon ({5})

4. Pentagram ({5/2})

5. Hexagon ({6})

6. Heptagon ({7})

7. Heptagram ({7/2})

8. Great heptagram ({7/3})

9. Octagon ({8})

10. Octagram ({8/3})

11. Nonagon ({9})

12. Nonagram ({9/2})

13. Great nonagram ({9/3})

14. Decagon ({10})

15. Decagram ({10/3})

16. Hendecagon ({11})

17. Hendecagram ({11/2})

18. Great hendecagram ({11/3})

19. Grand hendecagram ({11/4})

20. Great grand hendecagram ({11/5})

21. Dodecagon ({12})

22. Dodecagram ({12/5})

…

n. p-gon ({p})

…

### Skew polygons

Consider again the apeirogon. If, instead of all the corners lying in a single line, they lie in two parallel lines, L1 and L2, with the corners alternating between the two lines. If successive corners in L1, and successive corners in line L2, with the points on one line exactly midway between pairs of points on the other line, an infinite zig-zag is drawn. This line is both corner-transitive and edge-transitive, it is, by definition, regular.

This is a *skew apeirogon*.

In the same way that the original apeirogon was bent in an axis perpendicular to the apeirogon to form a polygon, a skew apeirogon can also be bent in an axis perpendicular to its plane. This forms the *skew polygons*.

### Zonogons

Whenever a polygon (not necessarily regular) has an even number of edges, with opposite edges being both parallel and equal in length, the polygon is called a zonogon, and each pair of edges represents a zone.

### Tesselations

A tesselation is formed by covering a plane with polygons. If the polygons are regular and equal, then it is a regular tesselation. If the polygons are all regular (not necessarily equal), and the tesselation is vertex-transitive, it is called uniform.

#### Regular tesselations

* Triangular tesselation **(36)**
* Square tesselation **(44)**
* Hexagonal tesselation **(63)**

#### Uniform tesselations

These tesselations can be *truncated*, i.e. the corners can be cut off and replaced with other polygons to a lesser or greater degree.

Starting out with the triangular tesselation, successive truncations give:

* *Triangular tesselation (36)*
* *truncated triangular tesselation (63)*
* Trihexagonal tesselation **(3.6)2**
* Truncated hexagonal tesselation **(3.122)**
* *Hexagonal tesselation (63)*

Starting out with the square tesselation,

* *Square tesselation (44)*
* Truncated square tesselation **(4.82)**
* *Square square tesselation (44)*
* *Truncated square tesselation (4.82)*
* *Square tesselation (44)*

Another way to form more uniform tesselations is by pulling the original polygons apart, and placing squares between the edges, along with polygons filling in the corners. This produces the rhombic tesselations.

* Rhombic triangular tesselation **(3.4.6.4)**
* *Rhombic square tesselation (44)*

These can also be truncated, giving:

* Truncated trihexagonal tesselation **(4.6.12)**
* *Truncated square square tesselation (4.82)*

Starting with the rhombic forms, and rotating the original polygons, the squares between the edges will become squat rhombi. At one point, the rhombi will be equivalent to pairs of equilateral triangles. When this happens, we have the snub forms.

* Snub hexagonal tesselation **(6.34)**
* Snub square tesselation **(4.3.4.32)**

### Non-Wythoffian tesselation

All the above transformations are known as Wythoffian. There is one further uniform tesselation that hasn’t been mentioned.

* Square triangular tesselation **(42.33)**

<= | Polytopes Wiki | Regular polyhedra =>