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Impossible Baseball

Image: Peter Miller
It has recently been brought to my attention that the official rules for Baseball have an impossible definition for the geometry of the Homeplate!
According to the official basesball rules 2.02, the Home Base should be a pentagonal slab of rubber conforming to the following specifications:
2.02 Home Base

Home base shall be marked by a five-sided slab of whitened rubber. It shall be a 17-inch square with two of the corners removed so that one edge is 17 inches long, two adjacent sides are 8½ inches and the remaining two sides are 12 inches and set at an angle to make a point. It shall be set in the ground with the point at the intersection of the lines extending from home base to first base and to third base; with the 17-inch edge facing the pitcher’s plate, and the two 12-inch edges coinciding with the first and third base lines. The top edges of home base shall be beveled and the base shall be fixed in the ground level with the ground surface. (See drawing D in Appendix 2.)


Following these instructions, we start with a 17” square:
From this we bisect two sides, chopping off two corners to meet at a point:
The shape of the triangle at the tip is thus this:
Using the Cosine Rule we can calculate the angle at the tip.
As we can see, this angle is ever so slightly greater than 90° This means it is impossible to comply with the rest of the rule to make the 12” edges coincide with the first and third base lines.
We can’t please all the constraints of the rules; something has to give. If we build the plate according to the rules, the edges will not be at 90°, and not point directly at the bases.

Alternative Solution

If, instead of flattening the tip angle, we stick religiously to the desire to keep the tip at 90° (to make these edges line up, perfectly, with the first and third base lines), we can do this by slightly adjusting the length of the biased sides.
Using a little Pythagoras, we can calculate the appropriate lengths:
If we make the biased sides a few thousands of an inch longer, we can satisfy the constraint that they point straight at first and third bases.

There’s another solution

If we keep the constraint of 12” for the bias sides, and 90° for the tip angle, we can make things good by slightly reducing the side of the base from 17”.
Again, from Pythagoras:
So, by reducing the length of the base from 17” to 16.971” we can also fix the issue.
Which compromise do you prefer?