Math Detours · Grade 5 · Around the corner · 28

Which letters are secretly the same?

Bend and stretch a wire letter without cutting or joining. Which letters can become each other, and what changes when squashing is allowed?

Name ____________________Date ____________

Tonight's idea

Use the exact wire letters drawn below. The thin line is the shape, and strokes join wherever they meet.

In the stretch game, you may bend and stretch wire, but never cut, join, or squash a whole branch to a point. If every point matches continuously to one point on the other shape, and the matching works backwards too, the shapes are homeomorphic. Ends, junction types and loops are preserved.

The squash game is looser. You may shrink dead-end branches away. If two shapes can continuously squash to the same smaller shape, they are homotopy equivalent. A tree can squash to a point, but a closed loop cannot disappear.

Stretch-same shapes are squash-same. Squash-same shapes need not be stretch-same.

These are two ways topology decides when shapes count as the same, without measuring lengths or angles.

Watch one: follow the wire

CL OD 2 ends · no loop 0 ends · 1 loop ↔↔
C stretches into L. O stretches into D. C cannot stretch into O because cutting or joining would be needed to change two ends into a loop.

To test C and L, start at one end and travel along the wire. There is one route to the other end, with no choice and no return loop. Bends and corners do not matter, so slowly straightening C and bending it into L gives a reversible change.

Your turn

ABCDEFGHIJKLM NOPQRSTUVWXYZ 0123456789
This is the wire font for every question. The 4 is open. A crossing is a junction.
  1. In the stretch game, sort C, I, L, S, Z, O, D, E, T and Y into secret-same groups. Use ends, junctions or loops to explain.
  2. In the squash game, sort all 26 letters into the no-loop, one-loop and two-loop groups. Listing the one-loop and two-loop groups is enough.
  3. F is stretch-same as E, T and Y. Start at the junction and explain why. Which parts may you bend or stretch?
  4. Sort the ten digits into squash groups by number of loops. Remember to use the open 4 drawn here.

Stretch: X against Y

Are X and Y stretch-same? Are they squash-same? Give a reason for each answer.

Answers

  1. C, I, L, S, Z are one group: each is a single route with two ends, no junction and no loop. O, D are one group: each is one closed loop with no ends or junctions. E, T, Y are one group: each has one three-way junction and three ends.
  2. One loop: A, D, O, P, Q, R. Two loops: B. Every other letter has no loop. Dead-end branches do not add loops and may be squashed away, so each no-loop letter shrinks to a point. Each one-loop letter shrinks to a circle; B shrinks to two circles joined at one point.
  3. From the one three-way junction in each letter, three unbranched routes lead to three ends. Corners and route lengths do not matter, so bend and stretch those three routes until any one of E, F, T and Y has the shape of another. Do not cut at the junction or join two routes.
  4. No loops: 1, 2, 3, 4, 5, 7. One loop: 0, 6, 9. Two loops: 8. The answer uses the exact drawings, so the open 4 has no loop.

Stretch. They are not stretch-same. X has one four-way junction and four ends; Y has one three-way junction and three ends. They are squash-same: both are trees with no loop, so all their branches can squash to a point.

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