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Sunday 28 April 2019

Fractional Solids

2 different versions of greater dodecahedrons.

Ball-shaped greater dodecahedron before stripping away the support material from the printer.

Noodling on cad with the shape, I never printed this one.  The circles in the centre were drawn small at first to help  draw construction axes, but in this pic I made them into a big feature.
Schematic of some of the shapes from the spreadsheet.

A few of the shapes.  Gumby is standing on a couple of partial development shapes, and is so pleased with himself he wants to run for prime minister of Australia.
View of 4, 5 and 6 sided faces.

Top view of 3 sided vertex with views below of (part of) equisided triangle, square and pentagon.

Raw Draftsight Plot.

During one of my last posts, I showed an application for fractional polygons. Hugh Duncan has detailed these shapes in a blog post, and  Wikepedia calls them star polygons.   Anyway, I got to thinking about them, and that there were oodles of them, but why couldn't there be oodles of solids based on them?  I only knew of 5 types of solid (4th pic down), maybe there could be more?

Anyway, I already did a spreadsheet for calculating the coordinates of a spiral based on fractional polygons (fractional polygons in this folder) and decided to adapt it to seed some fractional solid designs.  The tactic for  my shape hunting is to calculate the coordinates of a vertex of "n" regular polygons of "m" sides.

As an aid to calculate, I drew some vertices to scale in 2d and checked the calculated dimensions and angles against the drawings.

The pics above show the results sofar, including making some shapes, and plotting "solid seeds" in Draftsight using a script pasted from the spreadsheet.  I managed to make a polygon that was new to me but I later found is called a "great dodecahedron".  Anyway, I will keep exploring & report more on this later.  Not everything always joins up and I have made some fine on-screen messes!


Steve Nurse

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