TriMata

Hideki Tsuiki
Kyoto University

What kind of shapes will be created by connecting the following pieces through their connectors (i.e., concave and convex sides)?

We want to construct a shape without open connectors. Since it is impossible to form a shape without open connectors from finite number of pieces, let's build bigger and bigger clusters without deadlocks like the following.

Examples: large shapes:

There are "unexpected" connections that allow various shapes in addition. See the works by children at a workshop in the Pictures page.

Mathematical background :

Each piece consists of two kites obtained by dividing a regular tetrahedron into three equal pieces. Concave and convex sides are added in order to connect the pieces tightly. If one tries to create a shape by connecting convex to convex and concave to concave sides, one will rapidly reach a deadlock. Therefore, we will have the same solution sets whether we play it with concave and convex sides or flat sides.

The solution set is characterized by a (three-directional) triangular cellular automaton, which is equivalent to the rule 18 elementary cellular automaton. This is why we named this puzzle "TriMata" (and MitsuMata in Japanese which means three junction).

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