From: Don Reble <djr@nk.ca> To: math-fun@mailman.xmission.com Sent: Friday, September 23, 2016 1:52 PM Subject: Re: [math-fun] Bar bet

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"The volume of a regular tetrahedron (triangular pyramid with unit edges) is exactly half the volume of a square pyramid with unit edges." True or false? --rwg

Trick puzzle: prove that the volume of _every_ such pyramid (unit edges, regular polygon base) is a quadratic surd.

On 2016-09-23 13:59, Eugene Salamin via math-fun wrote:

The only remaining nontrivial case is the pentagonal base, and it's clear that all the needed heights are no more involved than stuff containing sqrt(5).

In[445]:= PolyhedronData[{"Pyramid", 5}, "Volume"] Out[445]= 1/24 (5 + Sqrt[5])

For the hexagonal base, the pyramid is flat, zero volume, and for higher order polygons, the lateral edges cannot reach the vertex while remaining of unit length.

-- Gene

In[587]:= PolyhedronData[{"Pyramid", 6}, "Volume"] During evaluation of In[587]:= PolyhedronData::notdef: PolyhedronData has no value associated with the specified argument(s). >> Why, do you suppose?-) --rwg