# Re: [Icehouse] Size for 4 - 6 pointers (Was: Giant Martian Coaster boards)

• FromCarl Worth <cworth@xxxxxxxxxx>
• DateWed, 10 Jan 2007 15:28:35 -0800
```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.

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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