## Diary: Kelvin’s Piecepack Pyramid Dimensions

Posted by kelvSYC on 8-26-2013

Almost as soon as I finished writing that last diary post, I really looked into the dimensions of my prospective piecepack pyramids, and whether I can get them 3D-printed. First, I want to investigate the dimensions.

The reference document dictates that the side of a piecepack pyramid form an isosceles triangle with a 36-degree angle at the tip of the pyramid for all sizes. As stated, the base of the pyramid range from 1/2″ (A) to 27/32″ (F), in roughly 1/16″ increments. (Pyramid F is slightly larger). My highly imprecise measurement method also reveals that the distance from the tip of the pyramid to a corner range from 13/16″ (A) to 1 3/8″ (F), in increments of roughly 0.3cm. Doing some math, the heights of these pyramids would be 0.73 inches (A) to 1.24 inches (F), with Pyramid E (at 1.13 inches) being “pawn-height” (a pawn is 1 1/8″ in height).

With a pyramid base increasing by 1/16″ from one size to the next, it would leave me with a thickness of 1/32″ for the pyramids (and allow the other 1/32″ for some “play”, similar to Looney Pyramids specs). The problem here is that 1/32″ is a tad too thin for most 3D printers that deal with plastic (a quick look at materials used in 3D printing reveals that most plastics require a thickness of 1mm; 1/32″ is roughly three quarters of that). If you are willing to print using metal and drive up your materials cost, it’s worth a shot though. But it’s likely to be cheaper to just cut sheet metal to make pyramids if you’re going that route… So, for the budget conscious (for the definition of “budget-conscious” that goes out of their way to make pyramids out of plastic, that is), it looks like 1/16″ is the smallest thickness that I can realistically use.

A 1/16″ thick pyramid would need 3/32″ difference between pyramid sizes if we were to go by the above. If we were to keep pyramid F at 27/32″, then pyramid A would have 3/8″ base. In other words, something slightly larger than a zero-pip Looney Pyramid (a Looney pawn is known as a “one-pip” pyramid, drone “two pips”, and queen “three pips”, their bases and heights form a natural arithmetic progression, so a “zero-pip pyramid”, created in practice by hacking off the tips of other pyramids, is a natural progression in reverse). If that’s too small and you would like to keep pyramid A at the half-inch square base, then pyramid F would be 31/32″ square – still permissible within the constraint that the base of pyramid F must be no larger than a quarter of a piecepack tile (ie. one inch square), and slightly smaller than a Looney queen.

Can I possibly make things even thicker for good measure? Possibly. If we had 3/32″ thick pyramids (ie. 1/8″ difference in base sizes), then pyramid A can have a 3/8″ square base and pyramid F can have a 1″ square base. All fairly good base sizes, and as 3/32″ is roughly 2.4mm, you could work with a wider choice of materials, I suppose. But 1/8″ thick bases is definitely out. I would imagine that a pyramid with a 1/8″ base (the base of A if E was 3/4″, like it is with the reference document) would be very difficult to handle.

So, now for the heights of said pyramids. The problem is that, the thicker we make our pyramids, the less likely that they will, in fact, stack neatly (that is, pyramid A should be completely obscured by pyramid B if you were to place it on top of pyramid A). While I haven’t tried out maintaining the specified heights and seeing if this occurs, I had been considering adopting the Looney Pyramid model of having a fixed height to base ratio. Specifically, Looney Pyramids maintains a 7:4 height-to-base ratio (up to 1/32″ of an inch), and if I were to take pyramid E to be “pawn height”, then a base of 7/8″ (used in the 3/32″ model and the 1/16″ model with the larger pyramids) would give these pyramids a height-to-base ratio of 9:7. Now a 9:7 ratio, rounded to the nearest 1/32 of an inch, almost exactly gives a height difference of 1/8″ between pyramid sizes. (Pyramid A, at 9/14″, would actually be closer to 21/32″ than 5/8″, but the heights for B, C, and D would fall between a nice eighth-inch multiple and the next 1/64″ larger, and for F the 1/64″ smaller) That seems a bit convenient, let’s see if this actually stacks…

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