Self-Tiling Tile Sets

These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.

Palingram

A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:

sallows palingram

This reminds me of a short short story by Fredric Brown:

THE END

Professor Jones had been working on time theory for many years.

“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”

Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.

“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”

Years many for theory time on working been had Jones Professor.

END THE

Borromean Tribars

borromean tribars

Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.

(Thanks, Lee.)

Pièce de Résistance

Take an ordinary magic square and replace its numbers with resistors of the same ohmic value. Now the set of resistors in each row, column, and diagonal will yield the same total resistance value when joined together end to end.

This paramagic square, by Lee Sallows, is similar — except that the resistors must be joined in parallel:

sallows paramagic square