Tetrads

Definition
A tetrad is a union of four (simply connected) shapes which are placed without overlapping in such a way that each shape shares some boundary (of positive measure) with each of the other three shapes².

Most important findings

No tetrad can be made out of four congruent convex regions (Karl Scherer 1979²).
Several examples exist (for both the symmetric and unsymmetric case) of curved tiles with only one vertex, four copies of which form a simply-connected tetrad (Karl Scherer 2005; see table below). This is the lowest number of vertices per tile known for this problem. One is also the absolute minimum of vertices per tile, because whenever three shapes meet at one point, at least one of the three boundary lines must have a vertex (cusp) at this point. Hence a simply-connected tetrad cannot be constructed from four tiles (congruent or not) which have smoothly rounded boundaries.
One can tile the plane with a simply-connected tetrad formed from congruent tiles; (Karl Scherer 2005; see table below)

Types of tetrads

tetrads where the four shapes join without gaps (i.e., simply connected tetrads).
tetrads consisting of four congruent shapes. Such a tetrad is called a congruent tetrad for short.
- simply connected tetrads consisting of four congruent shapes. This is a combination of the two conditions above. Some polysquare reptiles can form such simply connected congruent tetrads¹. There are also some polyiamond and polytan solutions to this mathematical puzzle. It is still an unsolved problem whether a simply connected congruent tetrad can be convex.
tetrads consisting of four similar shapes (i.e. figures of same shape, but with different sizes). Such a tetrad is called a similar tetrad for short. In contrast to congruent tetrads, a simply connected similar tetrad can be convex; even better: it can have the shape of a square (see table below).

Areas of interest

From the previous chapter we see that the following properties of tetrads are worth investigating:

Does the tetrad have gaps or is it simply connected ?
Are its four parts congruent/similar/neither ?
Are its four parts symmetric/assymetric ?
Are its four parts polysquares/polyiamonds/polytans/polyhexes/other ?
Can the four parts have curved outline / partly curved outline / fractal outline ?
- If the four parts are partially curved, how many vertices do the parts need at least? Can we find a solution for any given number of vertices? The table below shows examples of convex simply-connected tetrads with pieces which have any number of vertices apart from 2, 3, 5. Can one find solutions for these three missing cases?
Is the outline of the tetrad convex or not ?
- can the outline be a square ?
Is the tetrad symmetric or assymetric?
How many vertices does the outline of a tetrad have at least?
Do the tiles have to have straight lines as part of their outline?
Can the plane be tiled with tetrads formed from congruent (or similar) tiles? As we can see in the table below, for both questions the answer is yes. However, one could impose further conditions like symmetry of the tiles or number of vertices given.

Not all resulting mathematical problems have been solved yet. However, the examples in the following table will answer many of them.

Table of examples

The column V denotes the number of vertices per tile. Some tiles have obvious ways to generalise them. These extensions are what entries like 20+8n in the vertex column hint at.

Part 1: Congruent Tetrads With Gaps	V	Author	Comment
	6		9-ominoes; convex tetrad
	8		9-ominoes
	2		14-ominoes, symmetric pieces
	12		4-fold symmetric pieces
	0	Karl Scherer	4-fold symmetric tiles without vertices
	0	Karl Scherer	curved tiles without vertices
Part 2: Congruent Tetrads Without Gaps	V	Author	Comment
	10		27-ominoes
	32+8n	Frank Rubin, Karl Scherer	symmetric 24-ominoes, smallest symmetric polysquare solution
	20+8n	Karl Scherer	54-ominoes, symmetric pieces
	8		polysquares
	8		polysquares
	12	Karl Scherer	polysquares
	16	Karl Scherer	polysquares
	28	Karl Scherer	polysquares
	8	Karl Scherer	polysquare reptile; this tetrad tiles the plane; see below.
	8		polysquares, reptiles
	16	Karl Scherer	polysquares
	19+3n	Karl Scherer	polytans
	19+3n	Karl Scherer	polytans
partially curved, 16 vertices per tile	16+3n	Karl Scherer	partially curved
partially curved, 10 vertices per tile	10+3n	Karl Scherer	partially curved
partially curved, 8 vertices per tile	8+3n	Karl Scherer	partially curved
partially curved, 6 vertices per tile	6+3n	Karl Scherer	partially curved
1 vertex per tile	1+3n	Karl Scherer	curved tiles with only 1 vertex; tetrad outline with only 4 vertices.
1 vertex per tile	1+3n	Karl Scherer	curved tiles with only 1 vertex; tetrad outline with only 4 vertices.
1 vertex per tile	1+3n	Karl Scherer	curved tiles with only 1 vertex; tetrad outline with only 4 vertices.
1 vertex per tile	1+3n	Karl Scherer	curved tiles with only 1 vertex; tetrad outline with only 4 vertices.
1 vertex per tile, no straight parts	1	KS	No straight lines in the outlines of the pieces.
	27	Karl Scherer	polytans, symmetric pieces
1 vertex per symmetric tile	1	Karl Scherer	curved, symmetric pieces with only one vertex
	6	Frank Rubin?	polyiamonds
	7	Scott Kim	polyiamonds
		Karl Scherer	polyhexes
Part 3: Similar Tetrads With Gaps	V	Author	Comment
	6	Karl Scherer	polysquares reptiles
	6	Karl Scherer	polysquares, reptiles, square outline
	6	Karl Scherer	polysquares, reptiles, square outline
	6	Karl Scherer	polysquares, reptiles, square outline
Part 4: Similar Tetrads Without Gaps	V	Author	Comment
	6	Karl Scherer	polysquares, reptiles
	6	Karl Scherer	reptiles
Note that this similar tetrad not only consists of four reptiles, but is itself a reptile	6	Karl Scherer	polysquares, reptiles
	6	Karl Scherer	polysquares, reptiles
	6	Karl Scherer	polysquares, reptiles
Unique: a square, similar, simply connected tetrad	6	Karl Scherer	polysquares, reptiles
	6	Karl Scherer	polysquares, reptiles
	9	Karl Scherer	polyiamonds
Part 5: Maximum coverage	V	Author	Comment
	7	Karl Scherer	polyiamonds
	1	Karl Scherer	curved tiles
	19	Karl Scherer	polytans
	19	Karl Scherer	polytans; tiling the plane with tetrads (with gaps)
	16	Karl Scherer	polysquares; tiling the plane with tetrads (with gaps)
	8	Karl Scherer	polysquares; tiling the plane with tetrads (with small gaps)
	8	Karl Scherer	polysquares; tiling of the plane with congruent tetrads; no gaps