LEGO® Bour's Surface

Because The LEGO Company get paranoid about this sort of thing I suppose I'd better make it clear that I have no formal affiliation with them, that my views are my own and do not necessarily represent theirs, and so on. So if you think any of this is official you are as deluded as they are.

Yet another minimal surface - this is the biggest and most elaborate of all my mathematical LEGO sculptures so far. And although I still have a most of a tubful of yellow bricks left over, I only have six unused 1x3's.

As with most of my mathematical surfaces, I made use of some computer assistance. Just in case anyone's interested, here's the raw LDRAW .DAT file generated by my program for this sculpture. Beware - the .DAT file builds it out of 1x1 bricks. Actually constructing this out of larger bricks so that it holds together is a (non-trivial) exercise for the reader!

Bour's surface can be parametrised as

x = r cos(t) - r² cos(2t) / 2

y = -r sin(t) - r² sin(2t) / 2

z = (4/3) r^1/2 cos(3t/2)

and the model shows this parametrisation for 0 < t < 4p, 1 < r < 4. There's no particular mathematical reason for taking the image of an annulus like this rather than a disk, but the hole left in the center lets you see the structure better. And anyway, it makes building it more of a challenge and looks cool...

Here are some links to pages related to Bour's surface:

Bour's Surface - this page has a Java applet that lets you rotate an image of the surface in 3D with the mouse, and a reasonable description of what a minimal surface is.
MathWorld - Bour's Minimal Surfaces: actual equations, more Java applets and a list of fuether reading references here.

LEGO ® is a trademark of The LEGO Group, who have nothing to do with this or any of my other LEGO-related web pages.

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