Ingenious! It looks like an icosahedron is optimal amongst the Platonic solids in terms of 'how close it is to attaining isoperimetric equality':
import numpy as np da = 3 * np.sqrt(25 + 10 * np.sqrt(5)) dv = (15 + 7 * np.sqrt(5)) / 4 ia = 5 * np.sqrt(3) iv = (15 + 5 * np.sqrt(5)) / 12 (dv ** 2) / (da ** 3) 0.0066729900912859035 (iv ** 2) / (ia ** 3) 0.007328180769013705 1.0 / (36 * np.pi) 0.008841941282883075
(That is to say, the dodecahedron has a score of 0.00667, the icosahedron has a score of 0.00733, and the sphere has the optimal score of 0.00884. See the above code for exact closed-form expressions.)
Sent: Saturday, August 25, 2018 at 8:34 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] a huge boon to the dirigible industry
Speaking of lead balloons, Lowell Wood once computed that you could harvest a metallic asteroid by slicing it into 20 triangular slabs, welding it into an icosahedral vacuum balloon, easing it into atmospheric reentry, and bringing it to rest floating above the ground. Without internal pressurization with Moravec's niobium cannon balls. --rwg
On Fri, Aug 24, 2018 at 4:21 PM Bill Gosper <billgosper@gmail.com> wrote:
(Re-sent due to xmission.com braindamage.)---------- Forwarded message --------- From: Bill Gosper <billgosper@gmail.com> Date: Sat, Aug 18, 2018 at 9:36 AM Subject: a huge boon to the dirigible industry To: Wolfram Technical Support <support@wolfram.com>
In[31]:= ChemicalData["Nitrogen", "MolecularMass"]
Out[31]= Quantity[28.014, "AtomicMassUnit"]
(A mole of nitrogen gas weighs ~one ounce.)
In[32]:= ChemicalData["Air", "MolecularMass"]
Out[32]= Quantity[794.9, "AtomicMassUnit"]
Takes my breath away. --Bill
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