SOLUTIONS TO TOPOLOGY MAZE

Topological Mazes

DESIGNED BY Nick Rauh, Alice Peters, Mark Saul, Freddy Bendekgay
© 2018 Julia Robinson Mathematics Festival

Microfiber Maze

Problem II

Bend the cloth around so that:
  • Point A matches with point 4
  • Point B matches with point 3
  • Point C matches with point 2
  • Point D matches with point 1

You have formed a cylinder. We say we have changed the topology of the maze.

On this cylinder, you can connect two of the four shapes, but not the other pairs of shapes. Which pairs can you connect?

Solutions:
Circle to square Not possible
Circle to triangle Go through B/3
Circle to star Not possible
Square to triangle Not possible
Square to star Go through C/2
Triangle to star Not possible.

Note that certain answers imply other answers. The relation ‘accessible’ is symmetric and transitive.

Problem III

Now bend the cloth bent cloth so that:
  • Point E matches with point 10
  • Point G matches with point 8
  • . . .
  • . . .
  • Point J matches with point 5

And do the same problems on this topology. This is a different cylinder. Which shapes can you connect on this cylinder?

Solutions:
Circle to square Not possible
Circle to triangle Go through 7/H
Circle to star Go through 7/H, then 5/J
Square to triangle Not possible
Square to star Not possible
Triangle to star Go through J/5

Note that this topology isolates the square.

Problem IV

Imagine bending the cloth so that:
  • Point A matches with point 1
  • Point C matches with point 3
  • Point B matches with point 2
  • Point D matches with point 4

This surface is a Möbius Strip. Try connecting shapes on this surface.

Solutions:
Circle to square Not possible
Circle to triangle Not possible
Circle to star Go through B/2
Square to triangle Go through C/3
Square to star Not possible
Triangle to star Not possible

More Problems Using This Cloth

Problem V: Imagine bending the cloth so that: 
  • Point E matches with point 5
  • Point F matches with point 6
  • Point G matches with point 7
  • Point H matches with point 8
  • Point I matches with point 9
  • Point J matches with point 10

This surface is another Möbius Strip. Try connecting shapes on this surface.

Solutions:
Circle to square Not possible
Circle to triangle Go through 8/H
Circle to star Go through 10/J
Square to triangle Not possible
Square to star Not possible
Triangle to star Go through H/8, then 10/J

This topology isolates the square. Note that “circle to triangle” and “circle to star” implies “triangle to star”.

Problem VI: Imagine bending the cloth so that:
  • Point A matches with point 4
  • Point B matches with point 3
  • Point C matches with point 2
  • Point D matches with point 1
  • Point E matches with point 10
  • Point F matches with point 9
  • Point G matches with point 8
  • Point H matches with point 7
  • Point I matches with point 6
  • Point J matches with point 5

This surface is a torus (the surface of a doughnut or bagel). Try connection shapes on this surface.

Solutions:
Circle to square Go through 7H, then 5/J, then 2/C.
Circle to triangle Go through 7/H
Circle to star Go through 7/H, then 5/J
Square to triangle Go through C/2, then J/5
Square to star Go through C/2
Triangle to star Go through 5/J

Finally, every pair is connected!

This torus is created by ‘pasting’ the top (red) row to the bottom (yellow) row in the cloth, the pasting the left (blue) column to the right (green) column.

But what if we ‘pasted’ in the other order: blue to green, and then red to yellow? We’d also get a torus.

But the topology would be the same.  Just look at which letter end up matched with which numbers.

Problem VII: Imagine (and it takes some imagination!) bending the cloth so that:
  • Point A matches with point 4
  • Point B matches with point 3
  • Point C matches with point 2
  • Point D matches with point 1
  • Point E matches with point 5
  • Point F matches with point 6
  • Point G matches with point 7
  • Point H matches with point 8
  • Point I matches with point 9
  • Point J matches with point 10

This surface is called a Klein Bottle. It cannot exist in three dimensions(!). But we can imagine the connections. Try connecting shapes on this surface.

Solutions:
Circle to square Go through 10/J, then 2/C
Circle to triangle Go through B/3
Circle to star Go through 10/J
Square to triangle Go through C/2, then J/10, then 8/H
Square to star Go through C/2
Triangle to star Go through H/8 then 10/J

Note that any path that works for the cylinder in Problem II or the Möbius strip in problem IV also works here. Can you find other paths, not listed here, that also work?

Problem VIII: Now imagine that:
  • Point A matches with point 1
  • Point B matches with point 2
  • Point C matches with point 3
  • Point D matches with point 4
  • Point E matches with point 10
  • Point F matches with point 9
  • Point G matches with point 8
  • Point H matches with point 7
  • Point I matches with point 6
  • Point J matches with point 5

This surface is another Klein Bottle. Try connecting shapes on this surface.

Solutions:
Circle to square Go through 7/H, then 5/J, then 2/C
Circle to triangle Go through 7/H
Circle to star Go through B/2
Square to triangle Go through C/3
Square to star Go through C/3, then J/5
Triangle to star Go through 5/J

Which cylinder and Möbius strip gives solutions for this Klein bottle topology? What other paths can you find which connect the shapes?

Problem IX: Finally, imagine that:
  • Point A matches with point 1
  • Point B matches with point 2
  • Point C matches with point 3
  • Point D matches with point 4
  • Point E matches with point 5
  • Point F matches with point 6
  • Point G matches with point 7
  • Point H matches with point 8
  • Point I matches with point 9
  • Point J matches with point 10

This surface, which boggles the mind, is called a projective plane. Try connecting shapes on this surface.

Solutions:
Circle to square Go through 8/H, then 3/C
Circle to triangle Go through 8/H
Circle to star Go through B/2
Square to triangle Go through C/3
Square to star Go through C/3, then H/8,   then 10/J
Triangle to star Go through H/8, then 10/J

 Which of the previous surfaces gives guaranteed solutions here?