Math Detours · Grade 6 · Around the corner · 24

The eight ways to move a square

Pick up a square, do something to it, put it back in the same outline. There are exactly eight things you could have done, and they obey rules that reach all the way to particle physics.

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Tonight's idea

A symmetry of a shape is a move that leaves it looking exactly the same: lift it, turn or flip it, and set it back into its own outline. For a square there are eight. Four are rotations: by 0°, 90°, 180° and 270° (call them R0, R90, R180, R270, turning anticlockwise). Four are flips: over a vertical line (V), a horizontal line (H), and the two diagonals (D and D′). Try it with a paper square with an F drawn on it.

FR0FR90FR180FR270FVFHFDFD′
The same square after each of its eight symmetries. The dot marks the corner that started top-left. R turns anticlockwise; V and H are mirrors in a vertical and a horizontal line; D and D′ are mirrors in the two diagonals.

Do one symmetry and then another. The square is back in its outline, so the combined effect must itself be one of the eight. R90 then R90 is R180. V then V is R0, doing nothing. Mathematicians say the set is closed. Every move also has an undo: R90 is undone by R270, and every flip undoes itself.

Here is the table for the rotations alone. To combine two moves, find the row for the move you do first and the column for the move you do second. Example: row R90, column R90 gives R180.

then →R0R90R180R270
R0R0R90R180R270
R90R90R180R270R0
R180R180R270R0R90
R270R270R0R90R180

A collection of moves with closure, an undo for each, and a do-nothing move is called a group. The eight symmetries of a square are a group. So are the moves of a Rubik's cube, and the symmetries physicists use to classify particles. (R90 is also the matrix \(\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\) from the matrices sheet; try it in the matrix machine.)

Watch one: order matters

Label the corners of your square A, B, C, D clockwise from the top-left and track where the letters go.

ABCDR90→BCDAV→CBAD= D′ABCDV→BADCR90→ADCB= D
Corners A B C D clockwise from the top-left. Top row: R90 then V. Bottom row: V then R90. Same two moves, different result.

R90 then V ends with A and C swapped and B, D in place: the flip over the top-right to bottom-left diagonal, D′. V then R90 ends with B and D swapped and A, C in place: the other diagonal flip, D.

In ordinary arithmetic \(3 \times 5 = 5 \times 3\). In the group of a square, order matters. That is why a Rubik's cube is hard.

Your turn

  1. How many symmetries does each shape have? An equilateral triangle, a rectangle that is not a square, a regular hexagon, a circle. (Count rotations and flips separately, then add.)
  2. Using the labelled square, work out what single move equals each of these: V then V; H then H; V then H; H then V; R180 then V.
  3. Write down the undo of each of the eight moves.
  4. Explain why a flip followed by a flip must always be a rotation. (Think about the F: a flip turns it into a backwards F.)
  5. Copy the rotation table and extend it to all eight moves. There are 64 entries, and every row and every column must contain each of the eight moves exactly once. Use that rule to check yourself.
  6. Which moves can you make using only R90 repeatedly? Which can you make using only V and R90 together? What does that tell you about building the whole group from two moves?

Stretch: the cube

A cube can be rotated so that any of its 6 faces ends up on top. Once a face is on top, how many ways can the cube be turned while keeping that face on top? Multiply to count the rotations of a cube. (If you also allow mirror images, which you cannot do with a real cube, the count doubles.)

Harder: suppose a flat shape has exactly three symmetries. Explain why none of them can be a flip. (Hint: if V were one of them, then V then V is another, and so is what you get by combining V with the third move.)

Answers

  1. Triangle: 3 rotations and 3 flips, 6. Rectangle: 2 rotations (0° and 180°) and 2 flips, 4. Hexagon: 6 rotations and 6 flips, 12. Circle: infinitely many of each.
  2. V then V = R0. H then H = R0. V then H = R180. H then V = R180. R180 then V = H.
  3. R0 undoes itself. R90 and R270 undo each other. R180, V, H, D, D′ each undo themselves.
  4. A flip turns the F backwards; a second flip turns it forwards again. A move that leaves the F readable the right way round can only be a rotation.
  5. Full table, row first then column. Rows and columns in the order R0, R90, R180, R270, V, H, D, D′:
    R0: R0 R90 R180 R270 V H D D′
    R90: R90 R180 R270 R0 D′ D V H
    R180: R180 R270 R0 R90 H V D′ D
    R270: R270 R0 R90 R180 D D′ H V
    V: V D H D′ R0 R180 R90 R270
    H: H D′ V D R180 R0 R270 R90
    D: D H D′ V R270 R90 R0 R180
    D′: D′ V D H R90 R270 R180 R0
  6. R90 alone gives only the four rotations. V and R90 together give all eight (the four rotations, plus V, and R90 then V = D′, R180 then V = H, R270 then V = D). Two moves are enough to generate the whole group.

Stretch. With a chosen face on top there are 4 rotations about the vertical axis, so \(6 \times 4 = 24\) rotations of a cube (48 with mirror images). For the three-symmetry shape: if V were a symmetry, the moves would be R0, V and one more, X. Then V then X must be in the set and cannot be R0 (that would make X = V), cannot be V (that would make X = R0), so it is X, which forces V = R0. Contradiction. So all three are rotations, by 0°, 120° and 240°, like a three-bladed propeller.

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