Sierpinski Triangle

Clearly, a shape without open connectors is a collection of regular triangles. There are four ways to form a regular triangle by connecting three pieces. The following pictures show two of them, which we call Lions. The other two are called Mitsu-Mata, and are explained in the next page.

We call the three protrusions of a Lion connectors. The right and top connectors of the Lion on the left are facing each other. If we extend them to form Lions, then we have triangular shapes with a hole in the middle. There are three possibilities, and the following picture is one of them. This shape also has three connectors, and the right and the top connectors are facing each other. We can form a shape with a big hole and three surrounding holes by connecting similar shapes to these connectors. One can repeat it infinitely to form bigger and bigger shapes. We will call them level-n Sierpinski triangles (extended with three connectors), with a Lion level-0. (Sierpinski triangle is the limit of a similar procedure that generates finer and finer structures inside a fixed triangle. Therefore, it may be misleading to call this sequence that is getting bigger and bigger a sequence of approximations of the Sierpinski triangle.)

Our goal is to find an infinitely large shape without open connectors. Consider a sequence S₀ ⊆ S₁ ⊆ ...⊆ S_n ⊆ ... of level-n Sierpinski triangles and consider their union S. S almostly satisfies our requirement, but for the sequence we generated above, the leftmost connector is unchanged and it is left as an open connector of S. Therefore, one needs to connect something to this connector to obtain a shape without an open connector. One way is to form another sequence starting with forming a Lion on this connector. Another way is to connect three copies of them to a Mitsu-Mata (or its subspecies).

Here, we considered, as the following figure A shows, a level-1 approximation with the right and the top connectors facing each other. We have three possibilities of extending the red Lion to level-1 approximations, as the figures A, B, C shows. Here, the left and top connectors are facing each other in figures B and C, and the right connector is facing upwards in B and facing downwards in C.

A B C For each of them, next level approximations are obtained by extending the two connectors facing each other, and there are three possibilities of the arrangements of their new connectors. Therefore, by choosing an appropriate sequence, one can generate a sequence whose union does not have an open connector.

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