Lets say you did not believe in the beautiful platonic solids? What would be required for proof? Would you need to be convinced to derive the constructions for the proofs of the spreads between regular edges? I think that characteristic (no not angle) will be sufficient to give a strong comparison to the critical student.

Consider a regular quadrilateral. What are the spreads? how can you use the spread to to show proof of this?

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TopNewestOne can inherently construct Platonic solids using only the framework of rational trigonometry. I think there is a paper of Wildberger's that shows this, but I will check again.

For starters, consider an equilateral triangle in affine \(n\)-space and take the nedians of the triangle with parameter \(\frac{1}{3}\) (these are like medians, but they go through the points one-third of the way from one point to another, taken in cyclical fashion). Try computing all the quadrances & spreads associated with this construction, and see what you can do when you generalise this to projective space.

P.S. This was an exercise of mine very early on in my PhD, but I haven't had the time to revisit this problem.

P.P.S. I do not think it is wise to characterise Platonic solids purely by their spreads; while I could be wrong here (Wildberger's book is a reference for this), two of the Platonic solids share the same spread... I will edit this section if incorrect, but regardless characterising them in this way takes away a lot of the geometrical beauty of them that comes as a result of the construction I have explained above.

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What ways can be used to "show-off" this beauty(of Platonic solids)? Strictly algebra? Strictly geometric properties? Finite fields? Graph Theory? Arithmetic?

What would be the most student friendly way of being intrigued by these?

I think the main issue is developing a purposeful mindset for someone exploring such an abstract realm is key.

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