GEOMETRICAL APPLICATIONS

by Brian Bolt

The word 'Geometry' literally means 'Measuring the Earth', and the Euclidean Geometry with which most of us are familiar has its roots in surveying the flood plain of the River Nile. However, over the centuries it has been refined, formalized and abstracted to a point where it has little obvious contact with the 'real world'. It has been formed

into a structure based on axioms, theorems and riders remote from its origins. It is a very sophisticated subject which has only survived for so long because, until the middle of the 20th century most children did not receive any secondary education and had a mathematical diet which can best be

described as the arithmetic for book-keeping and the mensuration of simple solids. Algebra and Geometry were reserved for the privileged, very select minority, who received post-primary education. These subjects were not seen as useful in any practical sense but, rather like learning Latin, were believed to train the mind, sharpen the intellect and help in logical thinking. There was no expectation that the subject would be used or useful, more that it was part of our cultural heritage to which every intelligent person should be exposed.

A typical Geometry lesson in my experience as a pupil, in my early days as a mathematics teacher, and in the experience of my parents before me was for the teacher to go through the proof of a new theorem on the board which then had to be learned for homework to be tested the next day. Questions would then be set whose proof required the use of the newly acquired theorem. As pupils we learned the discipline of setting out our work with headings like:

'Given' 'To Prove' 'Construction' 'Proof'

and those of us who became mathematicians revelled in the

logical structure and tight arguments involved. The subject to us had a beauty of its own.

However, as teachers we soon discovered that the subject only made sense to a very small minority of our pupils. Yes, they might learn the proof of a theorem off by heart and get a good mark in the inevitable test. Some might even manage to use the theorem in proving further simple riders, but

'What is the point of this?'

was never far from their lips.

We live in a three-dimensional world, a world full of objects which interact with each other, which is a far cry from the geometry of points, lines and triangles common to most school textbooks. The purposes of education are many and varied, but at the centre must be the aim to help our pupils to a better understanding of the space around them, of the structures of the buildings, the bridges, the cranes, furniture and the thousands of machines and mechanisms

that influence their daily lives. There is a common fallacy that practical work in geometry is for young children and by the time they leave primary school they have outgrown this activity. But at all stages in learning about space and spatial relations there is no doubt that making and handling objects leads to questions and developments which would never arise by a person just sitting at a desk thinking. Gaudi is one of the prime exemplars of this. His models using string and weights were an essential aid to his design process for buildings such as the Cathedral of the Sagrada Familia. Issigonis, the renowned car designer was an inveterate model builder, and was given a large Meccano construction set for his retirement. Buckminster Fuller, the genius who invented the geodesic domes which now straddle so many sports stadiums throughout the world did much of his thinking through model making.

As teachers we will do well to get back to the real world as our starting point for many of our geometry lessons.

Maps, Plans and Surveys

It is taken for granted that people can understand and 'read' a map. Maps come in many scales and convey various kinds of information about space. Early explorers had produced very crude maps of the outlines of countries and their main features, such as Ptolemy's map of the British Isles made in about AD 150. Nowadays, we are used to very detailed maps: town plans for tourists and visitors; road maps for motorists; large scale maps for hikers; and topological maps showing train networks. Anyone visiting Barcelona for the first time will almost certainly have bought a map of the city showing its streets and main buildings so that they know where places are relative to one another. Drivers of delivery vans and lorries must be able to read maps and interpret all the symbols they use, while any car owner who ventures outside their immediate neighbourhood needs to be 'at home'

22 Mathematics in School, March 2001 The MA web site www.m-a.org.uk

Ivernia

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Albion

Map of the British Isles by Ptolemy

with maps. Perhaps you think this should be taught as Geography but it is essentially a geometric activity and links can usefully be forged between the two subjects.

Pupils need to be confident in being able to use a map to see how to get from one place to another, to have some idea how far they will travel, how often they will need to change

. . .

.... .....

u .

directions, what alternative routes may be possible, and so on. For hikers and cyclists the interpretation of the contour lines of the larger scale maps enable them to know what gradients to expect or avoid. Reading ready-made maps is a start, but producing one's own maps and plans leads directly into what data is required to fix points relative to one another in space. A good starting point is for pupils to draw a plan of the apartment, or the ground floor of the house in which they live. This demands an understanding of the layout of the

Cloak room

Living om I roomGarage

liall coal

Dining

e0 i 2 3 4 I I I I I

metres

rooms and passages, the need to measure accurately, to record the data, and a sensible choice of scale for the drawing. Such a plan has the advantage that the angles involved will normally all be right angles. The plan here is of the ground floor of a typical British semi-detached house.

Experience shows that this activity motivates pupils who will often want to go on to design their ideal home. It is a start. The next stage would ideally be to go out into the school grounds or park and survey a part of them. This is best achieved by having small groups of pupils working in teams and for the initial project not be be too ambitious. One useful strategy is to plant a number of flags in the ground and get the pupils to decide what measurements they need to make to enable them to make an accurate

scale drawing of their relative positions. Trundle wheels, or tapes or counting the number of paces of a member of a team can be used to measure distances.

D C

Bs

In this exercise the pupils will soon find the necessity to fix three flags at a time by measuring the lengths of the sides of the triangle they determine, and then add in other flags by considering triangles which have a side in common with what they have already drawn.

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With the four flags in the diagram a team might start by fixing triangle ABC then add in D by using triangle ADC. But D could have been placed by using triangle ADB or triangle BDC. A discussion of the different possible measurements which could be made to fix the four points is very enlightening. When angles, or directions are also brought into play the number of possible alternative approaches is surprisingly large, but they all help towards an understanding of what is required to fix points in space.

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Suppose a team based itself at A then it could draw lines from a point on a piece of paper in the directions ofB, C and D using a sighting rule, and then measure the distances AB, AC and AD. But the team could just as well have placed themselves at a point X inside the four flags, drawn lines in the directions ofA, B, C and D, then measured XA, XB, XC and XD to fix the four points.

Alternatively a team could walk along the sides of quadrilateral ABCD, measuring the length of each side as they go, and the angles at each vertex. These angles can be measured by using a sighting rule to draw the lines on a piece of paper from a vertex (flag) to the adjacent vertices

Mathematics in School, March 2001 The MA web site www.m-a.org.uk 23

and then using a protractor to measure the angle between the lines. Experience shows that the best way to record the measured data is directly on a sketch of the figure to be drawn.

S- veLo wel

In practice most areas to be surveyed do not have straight line boundaries; hedges, roads and streams often wriggle around. To map these a surveyor first approximates to them with straight lifies and then measures 'offsets' from the survey line to the required boundary. In effect he is setting up a Cartesian coordinate system. He measures how far along the survey line he has come and then how far away the boundary is from the line. Where the boundary is relatively straight few offsets are needed, but where it has many changes in direction more will be used.

The diagram which follows shows a survey of a small lake. The main survey lines form a triangle while the table shows the coordinates for each offset. For example, starting from A and walking towards B, the first offset comes after 14 m and is 10 m to D. The next offset is 25 m from A and is 3 m to E, and so on.

E U A; I

Po / 9 as ^'

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B C A

32 2 P 50 4 G 62. 4K t Z5 50

351 , F 87 2J I1 8N 7.5 3 E 282 l IM 14. IO O 20 6 H i L

a 8 a

Surveys can be linked to projects such as designing the layout of a temporary car park on an area of a school playing field. This could involve an investigation into the sizes of cars and their turning circles, and the best arrangement of the cars for ease of parking and getting out of the park. Not a congruent triangle in sight, but very much a spatial activity. For the more creative try designing a multi-storey car park ... then make a model of it!

Packing and Packaging

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We are all familiar with packages of different shapes and sizes from cuboids such as breakfast cereal packets, to triangular prisms of Toblerone chocolate, to cylindrical tins of fruit or soup, to plastic containers for liquids which come in exotic shapes. How efficient are these packages in their use of materials?

A telling investigation is to consider possible cuboids all with the same volume and then calculate the area of their

surfaces as a measure of the card they use. Take, for example, a volume of 96 cm3. This number has many factors so it is easy to find a significant number of different shaped cuboids with edges of integer length to minimize the arithmetic and emphasize the point of the exercise concerning the comparison of the surface area.

' g

12 cm x 4 cm 8 cm x 4 cm 8 cm x 6 cm 6 cm x 4 cm x 2 cm area x 3 cm area x 2 cm area x 4 cm area

160 cm2 136 cm2 152 cm2 128 cm2

A few of the possible cuboids are shown above. The longer and thinner they are the more card they use. For example, a cuboid measuring 48 cm x 2 cm x 1 cm has a surface area of 292 cm2 which is more than twice that of two of the ones

above, and so very wasteful of card. The most efficient

cuboid, of course, would be a cube with side 3x/96 which has a surface area of just under 126 cm2 so the cuboid measuring 6 cm x 4 cm x 4 cm is quite efficient.

This kind of investigation has all kinds of spin-offs. Mathematically, it involves pupils in the use of factors, volume and area, but it helps them to an appreciation of the best shape to use to conserve the packaging material and makes them more aware of the real world and conservation

in general. It could lead to questions about the possibility of more efficient shapes. For example, a prism with regular hexagonal cross-section can be designed with a volume of 96 cm3 and a surface area of only 120 cm2. Such a carton packs well in space but the container in which the cartons were packed would not be so convenient. The most efficient shape in terms of surface area to volume is a sphere but spheres don't pack to fill space so there would be inefficiencies in storing and transporting.

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Another similar investigation is to consider the volume of the open box which can be formed by cutting a square flap into each corner of a rectangular sheet of card and folding up the net formed. This was tried out recently with a group of 12-year-olds who worked in groups each starting with the same size sheet of card, 32 cm x 22 cm, and the members of a group all making different shaped boxes and computing their volume. The results are shown in the following table. The pupils were surprised at the result and wanted to see if they could achieve a higher volume by cutting out a 4.5 cm square. The investigation easily led into a graph which itself is an aspect of spatial relations.

24 Mathematics in School, March 2001 The MA web site www.m-a.org.uk

H (cm) W (cm) L (cm) Volume (cm3) 1 20 30 600

2 18 28 1008

3 16 26 1248

4 14 24 1344 5 12 22 1320

6 10 20 1200

7 8 18 1008 8 6 16 768

Polyhedra

The discussions on packaging naturally lead into work on polyhedra and this can develop in many ways. An investigation to find the twelve pentominoes can soon lead into different nets for an open cubical box, and it is a short step from that to find all the hexominoes and the subset of 11 of them which can fold into a cube. This last is a daunting task for most individuals but works well if tackled by groups of children combining their efforts and presenting their findings as a poster for the classroom wall with nets stuck on it which can be folded up to form cubes.

The six hexominoes above all fold to form a cube showing the variety of nets which are possible. But why not design a net for a cuboid or more interesting solids. The net for a tetrahedron is straightforward enough, consisting of just four equilateral triangles.

Regular tetrahedron

,,

But the shape which always motivates pupils is that of half a regular tetrahedron. It can often be found solid in pairs

made of plastic as a puzzle, where you are challenged to put the two pieces together to make the tetrahedron. Surprisingly most people find it very difficult to solve, so children like to make it and try it on their friends. Able pupils will be able to design their own net, others will probably best be served by printing the net on card and letting them cut it out. What is important is that each pupil feels successful and has the objects to handle and experiment with.

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6 cm

6cm 6cm

6 / 6m 6cm 6m os6cm

Taking the cube as a starting point many activities are possible. Consider, for example, how a cube may be cut to obtain a cross-section in the shape of: a square, a rectangle, a rhombus, an isosceles triangle, an equilateral triangle, a trapezium, a regular hexagon. These could be investigated by starting with a cube of Plasticene and cutting it with a knife, or by using a drawing and visualizing how it might be cut. The challenge would then be to create nets for the two

parts of the cube so that a permanent model could be made.

An investigation which has proved challenging with more able students has been to find original ways of dividing a cube in half, and then making the corresponding shapes. Four examples are shown above.

The formula for the volume of a pyramid is not the easiest concept to understand, but by making a model in which six identical pyramids fit together to form a cube with their points all meeting at the middle, a student has a practical way of understanding how the formula comes about. Such a model takes time to make, but it establishes the formula in a meaningful way. Another alternative model for the same purpose is one where three pyramids fit together to form a cube. In this case the pyramids are not symmetric and their points meet at one vertex of the cube as shown on page 26.

Mathematics in School, March 2001 The MA web site www.m-a.org.uk 25

There are a variety of challenging puzzles involving cubes. For example:

1. Show that it is possible to make a hole through a cube through which you can pass a larger cube.

2. Eight cubes are joined together to make a 2 x 2 x 2 cube. Show that the individual cubes can be rearranged to make the large cube either red or blue. That's easy, but now try the equivalent problem with 27 unit cubes which must be coloured in such a way that they can be put together to form either a red cube, or a blue cube, or

a yellow cube. How about N3 unit cubes to be coloured with N colours in a similar way!

3. There are numerous puzzles consisting of shapes to be fitted together to form a cube. They have probably all sprung from the interest shown in the puzzle invented by the Danish mathematician Piet Hein. It consists of seven shapes made from three or four unit cubes, as shown, which can be put together to form a 3 x 3 x 3 cube.

Games are also a way of getting children to think three- dimensionally. There is a three-dimensionally commercial version of O's and X's sold as Fours or Space Lines. It consists

of layers of perspex with a 4 x 4 pattern of holes in each layer. Two players take it in turn to put a coloured peg in one of the holes. The winner is the first player to obtain a line of four pegs of their colour. There are many possible winning lines because they can be horizontal, vertical or at an angle. One

diagonal line of pegs is shown in the diagram. How many possible winning lines are there? What is the smallest number of pegs a player can put on the grid to have at least one in every line? A similar game can be found where large wooden beads of two colours are placed in turn by two players onto a 4 x 4 array of vertical spikes. The strategies of playing this game are different as beads can only be placed at the lowest level or on top of existing beads. But the winner, as before, must get a line of four beads of his colour. The version

illustrated was made by drilling holes for dowel rods in a wooden base and using cotton reels for the pieces.

A Polyhedron Kit

A variety of educational manufacturers produce kits which consist of regular polygons of card or plastic designed in such a way that their sides all have the same length, and a means of being joined together. These are expensive to buy, but not difficult to make, and they have the advantage that they can be made from relatively small oddments of card using, for example, old Christmas cards or breakfast cereal cartons. The diagrams show how, by putting tabs on each side of the polygons, they may be fastened to each other using

26 Mathematics in School, March 2001 The MA web site www.m-a.org.uk

elastic bands, or for permanent models, they may be stapled. An efficient way of making a quantity of polygons is to first make templates which can easily be drawn around. The diagram suggests a common side length of 4 cm, but you may find a larger side appropriate with younger pupils. A large number of possible polyhedra which can be made using only squares, triangles and pentagons is shown, but experience shows that pupils show much more imagination than these may suggest and invent shapes of their own.

1/2 cm tabs

4 cm

4 cm

Rubber a band

Cube Octahedron Tetrahedron Cuboctahedron

Triangular prism

Triangular prism joined to tetrahedron

Square anti-prism

Snub cube

Square-based pyramid Icosahedron Dodecahedron Rhombicuboctahedrol

Mathematics in School, March 2001 The MA web site www.m-a.org.uk 27

Rigid Structures

Make a triangle linkage and quadrilateral linkage from plastic geostrips or card strips using paper fasteners to join their ends. If you place the linkages on a table and fix AB then the point C of the triangle linkage is also fixed, but points C and D of the quadrilateral linkage are free to move. This illustrates the innate rigidity of the triangle linkage, but lack of rigidity of the quadrilateral linkage. The traditional cycle frame makes use of the triangle's strength at the rear, but the main part of the frame BCDE, which supports the front forks is dependent on the strength of the welds at the joints. If you are observant you will not have to look far to see places where the rigidity of the triangle is used. Look at a folding chair, the fastening which holds a window open, the roof timbers in a house, an umbrella, a rotating clothes airer, the legs of an ironing board, the design of a traditional 5-bar gate, just to name a few.

B

D

A B

C D

E

A

B

28 Mathematics in School, March 2001 The MA web site www.m-a.org.uk

To increase understanding of which two-dimensional frameworks are intrinsically rigid an investigation can be made, for example, of what struts can be added to a quadrilateral in order for it to maintain one shape. It is soon clear that one of the diagonals will suffice but how about all of the other possibilities suggested by the following drawings. A traditional geometrical education gives little help in this and it is essential to make models to come to terms with the problem.

Having decided which of those frameworks are rigid the investigation can be followed up by considering pentagonal

or hexagonal frameworks. How about the following? Are any of them rigid? What is the minimum number of struts

required to ensure that an n-gon framework is rigid?

This concept of an intrinsically rigid framework should be extended into three-dimensions. There are so many structures in our modern society where they can be observed. Anywhere in Barcelona and district tower cranes can be seen like some exotic creatures looking down on the buildings being built beneath them. The structures of their towers and their jibs are prime examples of rigid frameworks as are the electricity pylons which stride like giants across the countryside. Many bridge structures such as the famous Forth Bridge in Scotland or the Sydney Harbour Bridge in Australia are well-known examples but so are the many railway bridges from the pioneering days when they were made of timber to the metal ones of today. The Eiffel Tower in Paris is such a structure, but on a smaller scale look at the scaffolding on a building site or visit a funfair to see many exciting examples.

Mathematics in School, March 2001 The MA web site www.m-a.org.uk 29

To investigate three-dimensional rigid structures one practical approach is to make use of drinking straws for the struts and joining them by using cotton, or preferably shirring elastic. The elastic is threaded through the straws to form linked triangles, pulling the elastic taut before tying a knot at a vertex. This method has advantages over others of joining straws using pipe cleaners or commercial plastic joiners which have an inbuilt rigidity thus masking the inherent rigidity of a structure. The simplest 3-D structure which can be built in this way is a tetrahedron, the main building block of many 3-D structures. It can be added to by building further tetrahedra onto one or more of its faces. The cube, although thought of as a rigid structure, only becomes rigid if a diagonal is added to each of its faces, and in so doing creating a polyhedron with triangular faces. - knot

knot 5

2 4

knot

No 3-D structure will be rigid unless it can be seen as an amalgamation of triangles. The square-based pyramid will look rigid as long as it sits on a table, but pick it up and it collapses as its bottom face is a square. But build another square-based pyramid as a reflection of the first in its base and you have an octahedron which is rigid. One attractive model to make is to start with an octahedron and

systematically add a regular tetrahedron to each of its faces. The resulting model can be seen as two large intersecting tetrahedra or a stellated octahedron depending on how it is displayed. Further, if the points of the stellations are seen to be at the vertices of a cube with the original octahedron having its vertices at the midpoints of the cube's faces.

Icosahedron

Many original structures can be made in this way, and it is instructive for pupils to make a study of structures in the real world and try to replicate them, whether a crane jib or a geodesic dome or a bridge.

This article on Practical Geometry is not intended to cover all the possibilities but to indicate an approach to the subject for the mainstream in a secondary school. Its aim is to increase pupils' awareness of the spatial world in which they live. Geometry taught as an abstract theoretical subject may appeal

Crane jib

Part of a

geodesic dome

greenhouse

to a few very able pupils but does nothing for the vast majority.

By considering the world in which we live our everyday lives as the starting point for spatial work there is no end to the possibilities to improve pupils understanding and motivation.

Acknowledgement Acknowledgement and thanks to CUP for permission to reproduce material in this article. The section on surveying includes diagrams from The School Mathematics Project Book I. Remaining diagrams are original for this article or reproduced from Mathematical Activities and Mathematics Meets Technology.

Further Reading The following resource books for teachers contain many appropriate ideas: Mathematical Activities More Mathematical Activities Even More Mathematical Activities

Mathematics Meets Technology 101 Mathematical Projects

These are all by Brian Bolt and published by Cambridge University Press.

Keywords: Geometry; Practical Applications.

Author

Brian Bolt, 31 Exe Vale Road, Countess Wear, Exeter EX2 6LE

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30 Mathematics in School, March 2001 The MA web site www.m-a.org.uk