Oops.
You are correct sir!
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In theory use an even number of slots +tabs and you have one shape.
In fact make one side and copy/paste/rotate 72 degrees.
Kerf adjust the resulting shape if you like.
I should build it to be sure but that seems right to me?
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I actually did that yesterday, and while it “should” work, it didn’t. The slots have to be extended beyond the original edge of the pentagon to put them on the same plane as the tabs, or, the tabs need to be able to bend, and they can’t with wood. (I quit playing with it when Marion did his magic because that’s the answer - you need different shaped tiles. Some for receiving and some for insertion.) Makes a great puzzle layout though. 
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Hold the phone…I might be incorrect here…I’m going to try one more thing utilizing the correct spacing for setting up a tab and the slot in the same plane…it might work.
Yeah I think it does work because here’s a photo that shows it doing so… yanked from earlier in the thread:
https://community.glowforge.com/uploads/short-url/n9uHQtd7aMtmZ0NzRYR1oMJZie3.jpeg
Pretty sure each side is the same here.
You need to use the law of sines to derive the tab length at the correct dihedral angle for a regular dodecahedron, but that’s easy enough to do.
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Yep, happy to report I was wrong…just hadn’t taken it far enough. The shape below works perfectly, with the exception of that little triangular bit at the 3 way corners.
dodec2.zip (1.2 KB)
You guys don’t even wait for the weeding though… 
(And it is kerf adjusted for medium DB, so it’s rather a tight force fit. Wiggle it.)
You can also just divide the length of one side of the original polygon into six equal segments and space your tabs/slots that way. It’s what I did and it worked just fine. It has to be six because the slot/tab combination is three - one section for slot or tab, two on either side for the mirror.
And picture of the unweeded result after quickly slapping it together:
(This was fun and I learned a bunch about how to think about spacing tabs from it.)
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It has to be greater than zero and even, 6 is arbitrary. The upper limit is derived by overall dimensions and laser properites (kerf and “burniness”) of your material. 2mm is a good baseline minimum kerf adjusted tab thickness for most hardwoods, but I wouldn’t recommend pushing it too far, lest you burn your precious (or yourself) up.
I found out there was a limit and stopped. @rbtdanforth went a bit further I think, and quantified how thin he could go. Am I remembering that correctly?
Anyway, if it were me, I’d build a side, then use clones of that one side to build my pentagons. That way I could adjust the tabs on the original path and modify all my pentagons in one shot. There’s nothing to stop you from using variable width or otherwise more decorative tabs (as these are just non-orthogonal finger joints), so there is a lot of room to play here.
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Interestingly I am also working on a rolling toy and was up 2 nights ago trying to figure out jointing after doing a test cut. I love math but I was trying a 2 to 1 tab ratio. I love this and really love this collaboration process in working things out here.
As for the designing I’ve learned a lot the last 2 months and am getting pretty good at illustrator and am now making my own files. I often copy and past elements together between files so give it a try. test cut at 1% power to align is my new trick!!!
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Ok. The latest is one built of clones. I haven’t joined them so they are still separate line segments.
Ok, this one is all clone. I left the original separate. Tweak that and all the hexagons change.
Do three tabs and three slots will work for all the tiles? I am not at my laser, so if someone were to try this in cardboard, they could answer the question.
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Yeah, that’s the shape. I tested it on DB up a couple posts so it should work.
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So one shape is all it takes to tile the whole dodecahedron? That’s fascinating. I wasted a lot of time last night.
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No you didn’t…you came up with a viable solution. That’s a win. 
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Going with the grain you have gone much smaller than myself, but any cut not with the grain is a real problem as they break super easy. In plywood this is not a problem as the grain is not crossed in a way that weakens it, though I suppose plywood with MDF would be a problem,
In plywood I cut to half the thickness for tab or gap but in full wood I limit it to the thickness of the material and even then I have to be careful.
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On thing about the odd number of tabs to slots, while complicating the whole build, it does make the corners sharper and more symmetric.
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oh my goodness. I know this is all going to be very very helpful and I can’t believe you put this together in a night. geesh. I am copying and pasting into my “really helpful inkscape/glowforge stuff” file I’m making. LOL. then will be the step by step seeing if I can learn it. thank you SOOOO much for this. this is much closer to what I am looking for and definately gets me in the right direction!
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this is way above my head…math! arrrgh
You don’t need to do the math, you can use a calculator like this:
https://www.analyzemath.com/Geometry_calculators/right_triangle_calculator.html
The dihedral angle of a dodecahedron is 116.57 degrees, *hand waving* your tabs should be about 1.118x as thick as your material.
It’s hard to explain without diagrams how you get there but basically we are case 3, we know side a (your thickness) and angle A (180-dihedral angle, so 63.43 degrees). Put your thickness in the side a field and 63.43 degrees in and hit calculate. You want the hypotenuse value.
Simple right? The easiest way to do this though is just to do slot length = thickness of material / sin(dihedral angle)
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Nice keyboard. Mechanical keyboard junky?
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Most definitely! 
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