# Hexagonal-triangular coil

Hexagonal-triangular coil
Rank2
Dimension4
Notation
Schläfli symbol${\displaystyle \{6\}\#\{3\}}$, ${\displaystyle \left\{{\frac {6}{1,2}}\right\}}$
Elements
Edges6
Vertices6
Related polytopes
ArmyTriddit
DualHexagonal-triangular coil
Convex hull6-2 step prism
Abstract & topological properties
Flag count12
Euler characteristic0
Schläfli type{6}
OrientableYes
Properties
Symmetry6-2 step prismatic symmetry
ConvexNo
Dimension vector(2,2)

The hexagonal-triangular coil is a regular skew polygon in 4-dimensional Euclidean space, which can be obtained by blending a hexagon and a triangle (${\displaystyle \{6\}\#\{3\}}$). It is one of five regular hexagons in Euclidean space.

Its edges form a subset of 6 edges of a 6-2 step prism, which is a variant of the more symmetrical triangular duotegum.

## Vertex coordinates

The vertex coordinates of the hexagonal-triangular coil are the same as those found in a 6-2 step prism.

## Related polytopes

The hexagonal-triangular coil is the second-smallest possible regular skew polygon in 4D Euclidean space, the smallest being the 5-sided pentagonal-pentagrammic coil.

It is appears as the face of ${\displaystyle \{6,3\}\#\{3\}}$ and ${\displaystyle \{3,6\}\#\{6\}}$, which are both 4-dimensional skew polyhedra.

### Other skew hexagons

The hexagonal-triangular coil is one of five regular hexagons in Euclidean space:

Hexagons in Euclidean space
Name Extended Schläfli symbol Dimensions
hexagon ${\displaystyle \left\{{\dfrac {6}{1}}\right\}}$ 2
hexagonal-triangular coil ${\displaystyle \left\{{\dfrac {6}{1,2}}\right\}}$ 4
skew hexagon ${\displaystyle \left\{{\dfrac {6}{1,3}}\right\}}$ 3
skew triangle ${\displaystyle \left\{{\dfrac {6}{2,3}}\right\}}$ 3
skew hexagonal-triangular coil ${\displaystyle \left\{{\dfrac {6}{1,2,3}}\right\}}$ 5