Thoughts from Cerulean:
Eeyore recently mentioned making a craft project out of paper pyramids, the biggest that can be made out of an 8.5"x11" paper sheet. That got me thinking, so I looked up the specs for Icehouse pyramids, and used Tony Vigil's PDF as a template. Of course I could just take one of those triangular configurations and blow it up to the limits of a sheet of paper, but then I remembered Eeyore's 5-pip pyramids. Why not make a whole-page pyramid that's to scale with the specs? I did some rough estimates, and found that two 8-point pyramids can fit on one page, side by side.
But the real discovery came next. As the spec shows, the two fundamental dimensions of pyramids, width and height, vary linearly with pip count. Observe...
- 1-pip: 9/16"
- 2-pip: 25/32"
- 3-pip: 1"
in increments of 7/32".
- 1-pip: 1"
- 2-pip: 1-3/8"
- 3-pip: 1-3/4"
in increments of 3/8".
Note that height here is measured to the vertex perpendicular to the base, not the distance up the side of a pyramid. Some trigonometry can get you that number; it's slightly longer than the basic height.
Working backwards with this linear relationship gives you A ZERO-PIP PYRAMID! That being
- Width = 11/32" (0.344", or 8.7mm for you European pyramaniacs out there)
- Height = 5/8" (0.625", or 15.9mm)
You could theoretically go one step further and make a negative-one-pip pyramid, but it's unrealistic to make a pyramid that is 1/4" tall and only 1/8" wide.
For general sizing of pyramids beyond 3-pips, these formulas will give you the sizes:
- (W = 11/32 + PIPS * 7/32)
- (H = 5/8 + PIPS * 3/8)
Again, note that this is the overall height, not the height as measured up the side of a pyramid. The side length H' would be
- H' = sqrt( H*H + W*W/4 )
As Eeyore points out on his 5-pointer page, the aspect ratio of Height to Width (H/W) is not constant. The larger a pyramid gets, the more squat it becomes, though by diminishing amounts as size increases.
The 0-pip pyramid has an interesting ramification for Zendo. I've once heard a Master say that the null koan is one that contains no pips. Though this is not the technical null koan definition, it has worked up to this point because all pyramids have a pip value. Now, with the 0-pointer, you can have a null koan full of 0-pip pyramids in all sorts of colors and orientations. Negative-pointers would needlessly complicate things further.
It should be noted here that the Giant Pyramids are an exception to this. Those pyramids simply take the standard 1-3 pip dimensions, and multiply them by 8. Thus the Giant Pyramids are different sizes than their pip-count-times-eight brethren. For example, a Giant pawn is 8" tall, whereas an 8-pip would be 3.625" tall. The Giant pawn is more comparable in size to a 20-pip pyramid (8.125" tall).
The real formula revealed?
After much discussion, and someone finally directly asked Andy to reveal the original formula, he told fans it was
The conclusion of Kit Cooper's memo is a set of recommendations for what he
thought we were after, and his calculations set the standard we now use. He writes:
"Of course, it is obvious that possibility (2),(C) is the correct choice. Not only do the pyramid Base Size and Face Height vary with a neat equation: BS = 4 / 7 FH = (4 + (2 ^ (PointValue - 1)) / 8, but the pyramids are similar too."
This, of course, means that pyramids do not get squatter as they get larger, but are similar. The actual pyramids are rounded to the nearest 1/32 inch.
It is worth pointing out that this formula does not give the correct dimensions for 1 and 2 point pieces. This would make 1 pointers 1/2 x 7/8, rather than 9/16 x 1, and 2 pointers 3/4 x 21/16, rather than 25/32 x 11/8. There is no rounding to be done. One could find values of a, b, and c, so that a formula of the form FH = a + b * c ^ PointValue would give results that round to the production values, but then we're back to inferring a formula from the production standards. Furthermore, the closer you approach the production values before rounding, the closer your extrapolated sizes will be to a linear progression.
If it is intended that the mids are similar, with an aspect ratio of 7/4, perhaps we should be taking the linear progression of the heights as canonical, and multiplying by 4/7 to get the base width. The resulting widths of 4/7, 11/56, and 1, do indeed round to the production values within the nearest 1/32 inch (indeed, to the nearest 1/64 inch).
Ultimately, the dimensions you choose currently boil down to a matter of personal preference.
Spotted in the wild
Though they don't conform to the linear size progression, Jesse Welton has produced a set of 0-pip pyramids by cutting the ends off of other pyramids:
Putting Them to Use
Like Homeworlds, most games could be played by replacing a set of 1-2-3 pyramids with 0-1-2 pyramids. Some games may require the players to add mentally one pip to each pyramid to maintain the scoring system. Some groups choose to alter the names of existing games slightly when played on a smaller scale--Baby Volcano, for example, or Miniature Pikemen.
Since zero-pip pyramids are not marketed products, they do not have a single official name. While the name, "zero-pip pyramid" is certainly clear in meaning, it is too long to be comfortably typed or spoken in conversation. In an 8 March 2007 letter to the Icehouse email mailing list, Paul Blake first coined the term "ZPIP" as an abbreviation for "Zero-Point Icehouse Pieces." This abbreviation became an instant hit with the mailing-list community.