On Wed, 10 Jan 2007 14:27:33 -0800, Christopher Hickman wrote: > Ok, so this is just completely wrong. The linear progression is > good for both the dimensions and the angles. A linear progression can be, but not all of them are. If we start with the "published" sizes from the following page: http://www.wunderland.com/icehouse/MakingIcehouse.html We can derive constant inter-size increments for height (3/8") and width (7/32") from those numbers. But if we do that, we do not get identical angles for each pyramid. That's easy enough to confirm by simply looking at the height/width ratio for each of those published sizes: Queen: 1.75 / 1.0 = 1.75 Drone: 1.375 / 0.78125 = 1.76 Pawn: 1.0 / 0.5625 = 1.777... (repeating) Symbolically, given the formulas you proposed for width and height: Width = (7x+11)/32 Height = (12x+20)/32 You can solve for the Height/Width ratio as: Height/Width = (12x+20)/(7x+11) So that's initially 20/11, (or 1.81...) for a size 0 pyramid but gradually approaches 12/7, (or 1.714285...) as x approaches infinity. Definitely not uniformly-shaped pyramids. Contrast this with Don's derivation of the size which is specifically designed to preserve uniform pyramid shape (identical angles) at all sizes. With his formulas: Width = (3x+5)/14 Height = (3x+5)/8 And here it's easy to see that the ratio is always constant: Height/Width = 14/8 = 1.75 To properly compare these two derivations, first note that the equations for height are identical, (they differ only by a factor 4/4). Then, notice that the difference in the width functions is really tiny, (less than 0.01"), in the range of sizes from 1 - 5 but increases outside that range. Of course, it would depend on the precision of the manufacturing process whether the difference is significant at any size. For a couple more data points, the difference isn't as large a 1/32" until a pyramid of size 10, and it's not until size 115 that it reaches 0.5", (and at that point we're talking about a pyramid that's 43.75" tall). > So I still think a "jumbo" set of 4-6 pointers would be sweet. There's definitely no doubt about that. ;-) -Carl
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