space-filling with solid polyominoes

Space-Filling with Solid PolyominoesAuthor(s): P. N. EbaSource: Mathematics in School, Vol. 8, No. 2 (Mar., 1979), pp. 2-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213437 .

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by P. N. Eba, Ecole Normale Superieure, North West Province, United Republic of Cameroon

Although much has been said and written about tessellations or the art of space-filling, and of polyominoes ever since S. W. Golomb coined the term in 1 953, the subject is still a fascinating one. Usually writers occupy themselves on filling a two-dimensional space with different types of polygonal cells, and then studying the pattern that emerges. My object here is to tessellate a three-dimensional space with solid polyominoes.

Solid or three-dimensional polyominoes are formed by gluing cubes together so that the two faces of adjacent cubes are wholly in contact. When this is done, we have just one monomino and one domino, two trominoes from three cubes and eight different tetrominoes from four cubes. We shall regard mirror images as different, since one cannot be superimposed on the other physically.

I shall consider in this article, only the tessellation of solid pentominoes, because the tessellation of the lower polyominoes can easily be inferred from that of pentominoes. There are in all 29 solid pentominoes, which for the purpose of easy identification, can be divided into three groups. We may call the first group, consisting of 12 pentominoes, the "Flats", because they can be made to lie flat on their sides, as shown in Figure 1. In this position, each of the flat pentomino is just one unit in height.

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Fig. 1 The Flat Pentominoes

(a) (b) (c) (d) (e)

(f) (9) (h) (i) (i)

(k) (I)



It should be noticed that the flat pentominoes are actually a three-dimensional model of the 1 2 plane pentominoes, where the squares have simply been replaced by cubes.

The rest of the solid pentominoes are not flat, since it is impossible to orient them in such a way that they can lie flat with height of only one unit. These non-flat pentominoes give rise to two groups. In one group, consisting of five pentominoes, each pentomino exhibits mirror symmetry. These are shown in Figure 2.

(a) (b) (C) (d) (e)

Fig. 2 The Symmetric Pentominoes

It should be noticed that symmetric pentominoes also exist among the flats; in fact, there are seven of them, six having mirror symmetry (a, e, f, g, h and i in Fig. 1) and one, Figure 1j, having rotational symmetry. But for the purpose of our classification, we shall call all of them simply "flats".

The remaining 12 pentominoes are skew - they are neither flat nor are they symmetric. They are however made up of six pairs, each member of a pair being the mirror image of the other. Like all mirror images, no member of a pair can be super-imposed on the other by a series of transformations. These six pairs of skew pentominoes are shown in Figure 3.

(a) (b) (C) (d) (e) (f)

(i) (i) (i) (i) (i) (i)

(ii) (ii) (ji) (ii) (ii) (ii)

Fig. 3 The Skew Pentominoes

We now come to the problem of tessellating these 29 solid pentominoes. Because of the nature of a cube, which is the basic "cell" in the pentominoes, we shall consider the tessellation of a three-dimensional orthogonal Cartesian space.

The simplest tessellation of solid pentominoes is obtained from pentominoes that can be grouped together to form rectangular blocks. These blocks can then be neatly stacked, like cases in a warehouse, and thus used in tessellating a 3-D space. These pentominoes are l a, 1b and 1 d, whose tessellation is shown in Figure 4. (The numbers la, ib, etc, used here and similar numbers used elsewhere in the article, refer to the pentomino shown in Fig. l a, 1 b, etc.)

(a)

(b)

(c) Fig. 4 Block Tessellations

For the rest of the flat pentominoes, members in each group interlock to form slabs, each of which can be extended ad infinitum, by adding more and more pentominoes. To tessellate the entire 3-D space, we pile one slab on top of the other, till the whole space is filled up. Figure 5 shows the tessellation of the remaining nine flat pentominoes.

(a) (b)

(c) (d)

Fig. 5 Slab Tessellation

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(e) (f)

(g9)

(h)

Fig. 5 cont. (i)

Each of the five symmetric pentominoes can be used for tessellation by interlocking them to form jagged- edged slabs, two units in thickness. These slabs could be extended ad infinitum by adding more and more chunks of pentominoes; and as in the case above, by piling one slab on top of the other, the entire 3-D space can be tessellated. This is shown in Figure 6.

Notice that the pentomino 2a interlock in pairs to form a shape similar to 1j; hence the tessellation of 2a is the same as the tessellation of 1j shown in Figure 5g. Similarly, the pentominoes 2d and 2e each interlock in pairs, as shown in Figure 6d, 6e, to form the "cross" shape similar to 1 i; hence their tessellation is the same as in Figure 5f.

For the tessellation of the skew pentominoes, we first note that since both members of the mirror pairs yield identical tessellations, we need consider only one member of each pair. We shall therefore consider only the tessellation of the pentominoes labelled (i) in Figure 3a.

Many of these pentominoes interlock in pairs to form shapes similar to the ones already considered. The pentomino 3a interlock as in Figure 7a to form the "L" shape, whose tessellation is the same as in Figure 5d. The pentomino 3b interlocks as in Figure 7b to form rectangular blocks, whose tessellation has been considered already. The pentomino 3d interlock in a similar way to form rectangular blocks as in Figure 7c.

(a)

Fig. 6 Tessellation of Symmetric Pentominoes

(b)

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(c) (d)

(e)

Fig. 6 cont.

(a)

(b) (c) (d)

(e) (f)

Fig. 7 Tessellation of Skew Pentominoes

The pentomino 3c interlocks in pairs to form the letter "T" shape, as in Figure 7d, and its tessellation is as in Figure 5b.

The skew pentominoes 3e and 3f are the most difficult to tessellate; fortunately they both yield identical tessellations. Let us start with the pentomino

Fig. 8 Tessellation of the Skew Pentominoes 3e and 3f

3e; we may link several such pentominoes together to form a continuous band, to which more and more pentominoes can be added. Now if we make two such bands, and then overturn one band so that it "lies on its head", as in Figure 7e, both bands will interlock, leaving no gaps in between. Several pairs of bands will interlock to form the slab in Figure 8.

Similarly, the pentomino 3f will form a band which interlock in pairs, as in Figure 7f, and several pairs will interlock to form the tessellation as in Figure 8.

We therefore see that we can use solid polyominoes, up to pentominoes, to tessellate a three-dimensional space, but we soon run into trouble with higher polyominoes. An obvious example is the solid heptomino with a hole in its centre, shown in Figure 9, which cannot be used to tessellate a 3-D space.

Fig. 9 A Heptomino with a hole

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space-filling with solid polyominoes

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