Sunday, November 23, 2008

Geeking Out

Trevor Blake is updating one of his articles, on how to build a 5/8ths geodesic dome out of 105 paper triangles, using just three distinct edge lengths and two kinds of triangles, a P type and an H type.

The Domebuilder's Handbook by John Prenis (1973) was influential, specifically the information on page 94 for building a 3-frequency icosahedrally based alternate, although we may differ in nomenclature, as I go with Classes I, II and III (what I learned down on the farm, as we say).

Anyway, I started with an rbf.Icosa object and scaled it by 1/radius for a radius of 1, meaning each of its E = 30 edges is slightly longer than 1. (V = 12, F = 20).

For center C, I added three vectors from the same triangle (X1, U1, Q1), and divided by 3, the resultant vector being said mid-point, at the intersection of perpendicular bisectors through each edge and opposite angle.

For other points I used scale factors of 1/3 along these same edges, did some vector subtraction and addition. These points then needed to be pushed out to touch the unit radius sphere, again by using radius reciprocals.

The resulting vectors were sufficient to check edge lengths and angles in the handbook. I'll do the open source thing and link to some code (my inhouse is a little different, sorry).

We did all this in Fine Grind, me using my Ubuntu Dell and Trevor occasionally snapping photos and wifiing them to me. Later, I posted more about our process to the Math Forum, with a link back to this post for more details, in case other design teams want to practice their paces.