maths resource: filling in space

Maths Resource: Filling in SpaceAuthor(s): Frank TapsonSource: Mathematics in School, Vol. 18, No. 2 (Mar., 1989), pp. 22-26Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216343 .

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by Frank Tapson, University of Exeter

Filling in Space Another 2-d spatial activity. We have previously looked at Dissections and Tangrams (Jan. 85 and Sept. 87 respectively) and now turn to Tessellations.

Tessellations 1 (MRes 105) is to make sure that the idea of a tessellation is understood, and that the pattern is seen and can be continued. The unit being used is the F-pentomino. The first two patterns on the sheet are easy, the third a little less so. The third one has an unusual characteristic for a tessellation, in that it can be coloured in using only 2 colours. Most tessellations need at least 3 colours, certainly the top two do if we want touching shapes to have different colours. A good supplementary question is to ask how many colours are needed for any particular tessellation. Of course it depends on what is meant by "needed". I feel that, if they are to be coloured in, then it should be done in a way that enhances the idea of the pattern, and not in some random manner just to make it look pretty. For instance, I think that each of the upper pair on the sheet needs 4 colours if the pattern is to be emphasised, by alternating, say, Red/Green in the upper row, and Blue/Yellow in the lower row for each tessellation. There are certainly 2 other ways of tessellating with this unit but they are not easy to find, and it is during this search that "holes" are likely to appear in the pattern! In all of this work pupils should be encouraged to make at least one template of the unit shape that can be moved around. It is not always easy to see just how a shape will fit in, especially if it needs to be turned over. (Have scissors and squared-paper of the appropriate size available.)

Tessellations 2 (MRes 106) takes things a little further. It is meant to stand on its own (i.e. not necessarily follow sheet No. 1), which is why the explanation at the top has been given, so as to make clear what we mean by a "different" tessellation. Also it gives the idea of using the primary unit to make a bigger secondary unit (in this case a square) which we know will tessellate. This partly answers the question often asked, "How many do I need to draw?" I try to answer that by pointing out that it is not a fixed number, but that there should be sufficient to demonstrate that a pattern does exist. At the very least the outer edges should match up so that we could imagine the whole thing being bent up and interlocking with itself. It is not hard to find 3 ways of tessellating with the given unit (the "Z" pentomino) - one way involves making 4 of them into a cross and using that as the secondary unit. The question posed at the bottom of the sheet usually proves to be a little more difficult. The primary unit has rotational symmetry only, and solutions are easy to find if it is not turned over. The start given involves two units of opposite handedness and the solution will certainly elude some pupils - until they have seen it!

Tessellations 3 (MRes 107) provides a variety of shapes that will act as units for making other tessellations on squared paper. If more shapes are needed, it is worth knowing that ANY shape made up of 3, 4, 5 or 6 squares (that is, the 2 trominoes, 5 tetrominoes, 12 pentominoes, or 36 hexominoes) will tessellate. So also will most of those made from 7 or 8 squares - but not all.

Finally MRes 108 has been provided as a master for making 7 mm squared paper. I prefer this size of square for tessellation work, and all the drawings on the other sheets have been made to fit that. 5 mm is a little bit "fiddley" to work with, and 10 mm squares make for rather spread-out work - though very good for display purposes. Since 7 mm squares are not commonly available I thought it might be worth including this sheet.

There is a lot of experience, as well as pleasure, to be gained from this work. If you have any pupils with some artistic ability, they could be encouraged to develop this in a very interesting way. This leads of course, into the sort of work done by the Dutch artist Maurits Escher (1898-1971). It might be useful to be able to refer to samples of his work. Two inexpensive works that would serve for this are -

The Graphic Work of M. C. Escher pub. by Pan/Ballantine The Magic Mirror of M. C. Escher by Bruno Ernst pub. by Ballantine Books N.Y.

Additionally -

Creating Escher-Type Drawings by Ranucci and Teeters pub. by Creative Publications does exactly what the title promises.

It is said that Escher did not do very well at school (except for art). One wonders how he might have fared if he had had maths lessons like this?

22 Mathematics in School, March 1989



TESSELLATIONS 1

Try extending each of these three tessellations by following the pattern which has already been started.

That is three DIFFERENT ways of tessellating with the SAME unit. That unit can also be used to make at least two other tessellating patterns. Try to find them.

MRes 105 x Mathematics in School, March 1989 23



TESSELLATIONS 2

Here we see two different ways of tessellating with the same unit shape.

The same unit could also be used to make a square, and it is easy to see how that could be repeated. So that is a third way of tessellating with it.

How many different ways can you find of tessellating with this unit?

MRes 106 x

24

Can you find a tessellation that starts off with two of the units placed like this?

Mathematics in School, March 1989



TESSELLATIONS 3

Each of these shapes can be used as the unit of a tessellation. Try it with some of them. See if you can find different ways of tessellating with the same unit. (The lines inside the shape are only there to show how it is made up from squares, or half-squares.)

A

B

C D E

F

G H

I

J

K

L

O

Q

M

N

P R S T

MRes 107 x Mathematics in School, March 1989 25



MRes 108 x

26 Mathematics in School, March 1989



maths resource: filling in space

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