The Cube and its Symmetries

Transcription

1 The Cube and its Symmetries This workshop is designed to show Y8/9 children how symmetry operations interrelate. We take the cube as our example. Administration In your area is a visualiser, an OHP and screen, and the single models you need to demonstrate. The 12 acetates are reproduced for printing at the end of these notes. The children work in pairs. At tables accessible to the pupils are pots of interlocking cubes (Multilink) and interlocking panels (Polydron Frameworks) and the prepared models each pair needs. Introduction In this workshop we shall do things but seem to do nothing. We shall perform symmetry operations, geometrical transformations which leave an object exactly as it was before. Our question is: In how many ways can we move something so that it looks and feels as if it hasn t moved at all? In other words, in how many ways can something fit its own space? This is the number of symmetries it has. There are 3 kinds of symmetry: 1. rotational 2. reflective 1. central But we shall see that they re all related. As Poincaré, a great mathematician of the last century, said, Mathematics is the art of giving the same name to different things. We start in 2-D, but then move to 3-D because we want to deal with the cube. 2-D symmetry: fitting an outline Acetates A1-6 E1 Teacher demonstration + student participation Taking as motif the Magic Mathworks smiley face, invite the children to compare and contrast the symmetries displayed by designs with: rotation centres orders 2, 3, 4; 1, 2, 3 lines of symmetry. 1. A1, A2, A3 have rotational symmetry of orders 2, 3, 4 respectively. 2. A4, A5, A6 have 1, 2, 3 mirror lines respectively. Necessarily, they also have rotational symmetry of orders 1, 2, 3 respectively. 3. Necessarily, A1, A3 also have central symmetry. For a rotation you move the acetate in its own plane.

2 For a reflection you can t because you re trying to swap the front and back of the picture with respect to the mirror line. You can however give the picture a half turn about the mirror line in the 3 rd dimension. Challenge the children to consider what this is going to mean when you come to 3-D objects. Challenge a volunteer to perform a half-turn rotation in the plane of the projector but using reflections. [Two perpendicular reflections are needed.] Point out the significance for future work that this angle is half the angle of rotation. Draw the children s attention to a special property of designs with rotation symmetry of even order: every image point corresponds to an object point in the direction of, and the same distance the other side of, a particular point, the centre. We can say it has been reflected in that point. We say the design has central symmetry. [On one of these designs mark the centre on a clear plastic strip. Mark a chosen point on a motif and give the strip a half turn. The point will come into coincidence with a matching point.] As above, challenge the children to consider what this is going to mean when you come to 3-D objects. Which of these symmetry elements are compatible, i.e. which symmetries can be found in the same design? Acetate A7 Take suggestions and complete A7 by ticking possible cells from the blanks. [The correct entries are shown with a P for possible.] Ask the children what rotation symmetry order 1 means. [No rotational symmetry. The object can only be given a full turn to bring it into coincidence with itself, i.e. it must be left alone.] Number of symmetry lines Order of rotation symmetry 1 P P 2 P P 3 P P 4 P P 5 P P 6 P P Can a design have rotation symmetry without mirror symmetry? [Yes: we ve seen examples in E1.]

3 3-D Symmetry: fitting a hole E2 Teacher demonstration Put models 1, 2, 3 in turn under the visualiser: Instead of a point about which rotations happen, we have a line. 2. Instead of a line in which reflections happen, we have a plane. 3. For central symmetry we still have a point. E3 Teacher demonstration Acetate A8 Project A8. How many rotation axes are there of each type? What is the order of rotation symmetry of each? How many symmetry planes are there of each type? 60 mm hollow E4 Pupil experiment perspex cube with pouring hole, Ask the children to half fill their cubes. x 15 Invite them to orientate the cubes so as to produce different cross-sections as displayed by the water surface. Measuring cylinder Invite them to achieve: with 108 ml line a square, marked, x 1 the largest possible rectangle, a regular hexagon; Jug as water and in each case identify the vertical symmetry planes supply, x 1 and rotation axes.

4 The red lines indicate the vertical symmetry planes, the white numbers the orders of rotation symmetry of the vertical axes

5 Destroying the cube s symmetry 8 blue interlocking E5 Pupil experiment cubes, x 15, stock of pink and Issue this challenge: green cubes to be raided 1. Use your blue cubes to make a cube 2 x 2 x 2. as required 2. Either by plucking individual cubes out of your model or by swapping red and also, if necessary, green cubes for blue cubes, try to remove all the cube s symmetries. The challenge is to do so by removing or swapping as few cubes as possible. [It is not possible to destroy the symmetry of a cube 2 x 2 x 2 by removing cubes though it is for one 3 x 3 x 3. Two additional colours are needed: 2 of one colour must be swapped along an edge, 1 of a second colour at a corner such that all 3 share one face. The point of the challenge is to show how easy it is to miss symmetry elements. 1. The order of rotation symmetry of the cube E5 Pupil experiment + Teacher demonstration Ask this question: How many ways can we turn a cube so that it fits the hole in which it sits (the order of rotation symmetry of the cube)? The children are to lay a hand on the table and make a right angle between thumb and forefinger. This will be the hole in which they are going to fit their cube. Use your demonstration cubes (below) to perform the experiments, which the children repeat with their unmarked, wooden cubes. 40 mm 1. Ask the children to choose one of the cube s faces beechwood to lie on a matching face in the imaginary cube. cube, x 15 Show that the cube can be turned 4 times to fit. 2. Ask them to choose one of the cube s vertices to lie in a matching vertex of the cubic hole (the junction of thumb and forefinger). Show that the cube can be turned 3 times to fit. 3. Ask them to choose one of the cube s edges to lie in a matching edge of the corner cube (the thumb). Show that the cube can be turned 2 times to fit.

6 Acetate A9 Announce that they re now going to compute the same number in 3 different ways. Complete A9 together: We can stand the cube any of 4 6 faces. The order 4 x 6= ways of round on 3 8 vertices. symmetry 3x 8= any of is its 2 12 edges. therefore 2x 12= The cube s reflective symmetries 8-colour E6 Pupil experiment cube, viz. 4-colour The picture below shows the 8-colour cube to the left of the cube with semi-silvered mirror, in which the reflection can be seen. opposite The children must use the loose cubes to build the mirror form vertices and bring it into coincidence with its reflection. distinguished, x 15 Tell them that the mirror form is called an enantiomorph. The same 8 cubes loose, x 15 Semi-silvered mirror unit, x 15 2 cubes, red, blue, green, yellow, x 2, x 15 Ask them if they can turn the two cubes so that they look the same. [Answer: No: they are not superimposable.] E7 Pupil experiment Exhibit D, E, F under the visualiser for the children to build. They predict which they think will be superimposable, build the reflected forms, then sit object and image alongside to check as in E6. [All are superimposable. D and E have a mirror plane so it s clear that the parts either side can take the roll of front and back and be swapped. More interesting is case F, which we come to later.]

7 D E F What happens when you reflect something in a mirror? Ask them to recall what happens in 2-D: you reverse front and back to get the reflected form by turning the picture over. This involves a rotation in the 3 rd dimension about the mirror line. How can you perform a rotation in the 4 th dimension about the mirror plane? [Answer: by turning the cube inside out.]

8 Promise the children that they will shortly do that by building a hollow cube but that they can immediately swap front and back of their 8-colour cubes. As for E6 E8 Pupil experiment The children confirm that the swapping operation works. Point out two possibilities: (a) They can swap a block of 4 cubes across a plane parallel to a face. (b) Thinking of the diagonal plane in E, they can swap, for example, the orange-+-blue tower for the green-+-yellow. Polydron Framework interlocking squares, as needed Acetate A10 E9 Pupil experiment Tell the children the time has come to turn things inside out. They are to check whether or not a left-hand glove is superimposable. Show A10 and ask the children to perform the experiment. The connection between rotations and reflections Demountable kaleidoscope, x 15 E10 Pupil experiment Give the children a little time to explore the kaleidoscope. The children should insert a hand with thumb extended, a nonsuperimposable object therefore, between the two mirrors, and alter the dihedral angle. Ask them to comment on the first reflections in the two mirrors.

9 [The two images are the opposite way round to the object, and therefore the same way round. The angle between them is twice the angle between the mirrors.] Point out that what we noticed in the experiment with the smiley face design with order 2 rotational symmetry: that reflection in two perpendicular mirrors gives a half-turn, is just a special case of the general statement: reflection in two mirrors set at a certain angle is equivalent to a rotation of twice this angle about their line of intersection. Tell the children the significance of this: we can perform all symmetry operations by means of reflections alone. The cube s central symmetry Exhibits as described, x 1 E11 Teacher demonstration + Pupil experiment Present under the visualiser ff. models in turn: (a) to (c) are models with only central symmetry. (a) Model 3 again. Remind the children what central symmetry entails. (b) An open rhombohedron with space diagonals in cotton. Point out how the space diagonals join matching points as the transparent strip did in our 2-D experiment. (c) A calcite crystal as a natural example of such a rhombohedron. (d) to (f) demonstrate that solids with central symmetry consist of halves which are themselves enantiomorphs, but, if one half is reflected in 3 perpendicular mirrors and fitted to the other, a superimposable solid results. Semi-silvered mirror, 2 cubes of red, blue, green, yellow, x 15 (d) Model F: our 4-colour cube. Point out that the colouring means the cube has only central symmetry. Remind the children that they have shown in E7 that it is nevertheless superimposable. (e) Two copies of model 3. Orientate them first as mirror images, but then give one a half-turn perpendicular to the imaginary mirror so that it ends up with the same orientation as the other. (f) The children are to make mirror forms as shown. Demonstrate with the visualiser that these are enantiomorphs. The children must then try to fit them together to make their 4-colour cube. Acetate A11 Project A11.

10 The children may note: 1.) After 2 perpendicular reflections, the original has made a half-turn. 2.) After a 3rd reflection, it has its mirror form. 3.) But, in support of E11 (f), it is then in its correct orientation to fit the original and make a complete 4-colour cube. Polydron Framework Interlocking squares, as required Acetate A12 E12 Pupil experiment Now the children are to build their original 4-colour cube, but in a hollow version, and turn that inside out. They must build two copies, one to remain unchanged. Ask them to start with the trade mark Polydron on the inside. They ll know if they ve succeeded in turning their model inside out if it ends up on the outside. Show A12. Ask them how to get from frame 4 to frame 5. The answer is to give the mirror form a half-turn about an axis perpendicular to the mirror, shown by the blue line. [Demonstrate with the visualiser.] Point out to the children how, to take one space diagonal as an example, the red diagonal ends up parallel to the original direction. Tell them that the reason the inside-out form is just the same as the original is that they ve effectively reflected the cube in a point and, the cube having central symmetry, this has made no difference. They ve swapped one end of a space diagonal fort the opposite end, but these are identical.

11 Ask the children whether or not the 4-colour cube is superimposable. [They have demonstrated that it is in E7 and again here.] Ask them if the 4-colour cube has a plane of symmetry. [It does not.] Ask them what has to be true of a solid for it to be superimposable. [It needs either a plane of symmetry or a centre of symmetry or like the plain cube, represented by the perspex cube used in E3 or the beechwood cube used in E5 - both.] All the cube s symmetries by reflection E13 Pupil experiment Demountable The aim is to realise the image of a 2 x 2 x 2 cube. kaleidoscope, extra mirrors 1. Each pair simply arranges accordingly 8 cubes on the table. added as 2. They then set their kaleidoscope at a straight angle and arrange 4 cubes so that, combined with described, the reflection, they have their larger cube. x They now set it at a right angle, fitting a stack of 2 cubes to realise the same overall image. 4. Now give them a mirror to set horizontally beneath their kaleidoscope, (thus converting their 20 mm dihedral kaleidoscope into apolyhedral one). All they now need do is set 1 cube in the corner. Plastazote 5. The next step is to reduce the dihedral angle to 45. Give each group a half-cube, sliced by a cubes, symmetry plane containing a pair of opposite edges. whole or 6. Finally, swap the horizontal mirror for one designed to fit the 45 dihedral kaleidoscope at dissected, as needed 45 to the horizontal. What you now give them is the previous piece sliced by a further, corresponding cut to produce an orthoscheme. This is a tetrahedron with the same height as the preceding prism and therefore with 1/3 of its volume. Ask the children to calculate how many wooden pieces of the last kind make up a complete 2 x 2 x 2 cube. [Proceeding through the steps, we have: (2.) 1 2 x (3.) 1 2 x (4.) 1 2 x (5.) 1 2 x (6.) 1 3 = 1 48.] Point out to the children that we could have taken any one of 48 such pieces and fitted it into a kaleidoscope like ours to create the whole. We can therefore say that the order of the full symmetry group of the cube is 48.

12 Blue lines mark the edges of the original kaleidoscope, red lines the edges of the third mirror and the lines in which it cuts the original. Successive dissections of a single cube Kaleidoscope E14 Pupil experiment as for E13, Polydron We pass straight to step 4 in E13, using Polydron Frameworks in Frameworks: place of Plastazote for our single cube and its dissections. However, we squares: omit step 5 and go straight to step 6. 8 blue, right-angled isosceles triangles: 10 blue, 2 yellow, x 15

13

14 The link between the 24 rotational symmetries and the 48 reflectional symmetries Ask the children to recall from E5 what the order of rotational symmetry of the cube is. [24] Ask them what the connection is between that number and the order of reflectional symmetry they ve now found. [The second is twice the first.] Ask them why they think that is. [For each position achieved by rotation, they could turn the cube inside out, giving 2 reflections for every 1 rotation.]

15 A1

16 A2

17 A3

18 A4

19 A5

20 A6

21 A7 Number of mirror lines Order of rotation symmetry

22 A8

23 A9 We can stand the cube any of ways round on any of its faces. The order X = vertices. of symmetry x = edges. is therefore x =

24 A

25 A11

26 A