I enjoy making tessellations in GeoGebra that have edge alterations, in the style of M C Escher. I recently renoticed a print of his which featured an interesting double spiral tessellation.

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By collecting data in GeoGebra (http://www.geogebratube.org/material/show/id/32449) I became convinced that it's essentially a logarithmic spiral of right isosceles triangles (The post with those images is immediately after the previous post on my tumblr.)

But I'm having difficulty describing the edge alterations geometrically. Is it dilation of midpoint rotations? I'd be interested in how other people see the spiral or the tessellation transformations.

  • 1
    $\begingroup$ At first glance I'd say this looks like an iterated loxodromic Möbius transformation with the two vortices as fixed points. But after a quick experiment I have to concede that this first impression was wrong. $\endgroup$
    – MvG
    Mar 16, 2013 at 15:47

1 Answer 1


The light and dark blue logarithmic spirals are suitably translated and scaled versions of $$ r = e^{0.7\theta/(2\pi)} $$ in polar coordinates $(r, \theta)$.

Escher's fish with superimposed logarithmic spirals

To confirm the OP's suspicions and MvG's comment, Escher appears to have hand-interpolated near the center. It appears there is no conformal transformation sending the spirals to loxodromes, compare the following image of loxodromes under a Möbius transformation:

A pair of loxodromes


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