I was looking at this neat page on logarithmic spiral tilings when a question popped up:


It seems that in all of the tilings shown, the area of each tile is exponentially increasing as a function of the distance to the origin. Are there any radial- or spiral-type tilings (or "tiling-like" configurations) in which the area of each tile is a polynomial function of the distance to the origin, say $r^{1/2}$ or $r^2$? I don't require similarity of shapes or anything, just a simple way to fill the plane with shapes possessing (or approximately possessing) this property. I guess it would be nice if the shapes stay reasonable rounded and convex (i.e. not really long and skinny).

  • $\begingroup$ Obviously, one can do a tiling with parallelograms (or triangle, or anything one can make a parallelogram from) and any rate of growth. But I do not see any other shapes. $\endgroup$ – ε-δ Dec 10 '11 at 21:30

            Grunbaum and Shephard book cover
See Section 9.5, "Spiral Tilings," p.512ff. These remarkable tilings go back to H. Voderberg in the 1930's.

  • $\begingroup$ In these tilings the size does not grow... $\endgroup$ – Anton Petrunin Dec 11 '11 at 3:25
  • $\begingroup$ Yes, my point was that exponential growth in the OP's examples is not essential. And I guess $f(r)=1$ is technically polynomial growth! :-) $\endgroup$ – Joseph O'Rourke Dec 11 '11 at 14:10
  • $\begingroup$ No, $f(r)=\exp(cr)$ with $c=0$ is exponential growth :) $\endgroup$ – Piero D'Ancona Dec 11 '11 at 16:17
  • $\begingroup$ @Piero: Ha! Touché! $\endgroup$ – Joseph O'Rourke Dec 11 '11 at 17:22
  • 1
    $\begingroup$ The cover of this book is frequently mistaken as a Voderberg tiling. In fact, it it not. The Voderberg tile is shown clearly in Fig. 3.2.4 and the tiling is in Fig. 9.5.1. $\endgroup$ – Cye Waldman Dec 22 '17 at 17:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.