Hi Phi Fans,

Some time ago the Harriss Spiral created some interest, so i thought i may try another shape for members to look at using geometry, link to Harriss Spiril below:-

https://www.geogebra.org/m/sdTJfqb7 We know that the Pentagon, Pentagram, decagram, decagon, and dodecahedron are all related and fit together like a glove, but only by using ten points in a circle not five.

The 3D dodecahedron appears by where the lines cross in lower diagram:-

https://en.wikipedia.org/wiki/Decagon#/File-9-simplex_t0.svgScroll down above link to Petrie Polygons and tap on 9-Simplex to make larger.

However this will show two opposite pentagons etc. etc.

If we can double up, lets try the Golden Spiral and mirror image, the shape that would give is a Cardioid, be careful i know what you may be thinking, let me finish my observations first, i have to show how i fit all the other shapes into the Phi spiral and mirror image, it turned my brain into custard for years, but now i think i can do it, i'll continue soon slowly, i'm getting better after breaking my spine! Link to Golden Spiral below:-

https://en.wikipedia.org/wiki/Golden_spiral Use a face mirror under the spiral to see Cardioid.