Further Questions about Tetrads
15. Which tetrads can be made with hexagons?

The smallest n-gons which can be assembled to a tetrad with no hole are hexagons (n = 6).
(The above holds if the conjecture that holeless pentagons cannot form a tetrad is true.)
What is the smallest possible number of vertices VT of a tetrad made of hexagon tiles?.
Polyomino tetrad with VT = 8

Suitable polyomino tiles have at least 6 edges and 6 vertices.
Although Scott Kim's polyomino tetrad has a hole, the number of vertices is very small.
Moreover the shape of the tetrad is mirror symmetric.

Polyiamond tetrad with VT = 8

Suitable polyiamond tiles have at least 6 edges and 6 vertices.
The marvelous tetrad of Scott Kim and Frank Rubin has only 8 vertices.
But also a polyiamond tetrad with a hole can have only 8 vertices.


Open problem: Is there a tetrad which is a triangle and has a triangle hole (and no other holes)?

Polygon tetrads with VT = 9 and VT = 7

In Scientific American another mirror symmetric tetrad of Scott Kim was published. It has 9 vertices.

This tetrad can be modified. And we obtain a new record: VT = 7
The tetrad is a heptagon

Construction of this tetrad
In order to get a balanced shape we add the condition c = d.

Open problem: Is there a holeless tetrad which is a hexagon?

Other polygon tetrads

Two tiles form a square. VT = 12.

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 Walter Trump, Nürnberg, Germany, ms(at)trump.de, © 2020-02-12 (last modified: 2020-02-12)