# Hat

Hat
Rank2
TypeEinstein
Elements
Edges14 (all distinct)
Vertices14 (all distinct)
Measures (smallest edge length = 1)
Perimeter${\displaystyle 8+6{\sqrt {3}}}$
Abstract & topological properties
Flag count28
OrientableYes
Properties
SymmetryI×I, order 1
Flag orbits28
ConvexNo
Net count7
History
Discovered by
• David Smith
• Joseph Samuel Myers
• Craig S. Kaplan
• Chaim Goodman-Strauss
First discovered2023

The hat is a 14-sided polykite, which along with it's mirror image can tessellate the plane, but only non-periodically. This makes it a solution to some formulations of the einstein problem. It can be formed as an outer-blend of eight 60°-90°-120°-90° kites, or 4 mirror-symmetric pentagons.

Two of its sides meet at an angle of π , making them appear as a single edge. However there must be a vertex at that location for the aperiodic tiling to work.

## Vertex coordinates

Vertex coordinates for a hat with smaller edge length 1 can be given by:

• ${\displaystyle (0,0)}$,
• ${\displaystyle (0,-1)}$,
• ${\displaystyle (0,3)}$,
• ${\displaystyle (\pm {\sqrt {3}},0)}$,
• ${\displaystyle ({\sqrt {3}},\pm 2)}$,
• ${\displaystyle (-{\sqrt {3}},2)}$,
• ${\displaystyle ({\sqrt {3}},3)}$,
• ${\displaystyle \pm ({\frac {\sqrt {3}}{2}},-{\frac {3}{2}})}$,
• ${\displaystyle ({\frac {3{\sqrt {3}}}{2}},\pm {\frac {3}{2}})}$,
• ${\displaystyle (-{\frac {3{\sqrt {3}}}{2}},{\frac {3}{2}})}$.