On 10/21/08, James Buddenhagen <jbuddenh@gmail.com> wrote:
Here is a specific example of a tetrahedron which is continuously deformed with volume changing but surface area remaining constant.
As 'a' varies from 0 to sqrt(2), the tetrahedron with vertices (-a,-a,1), (a,a,1), (b,-b,-1), (-b,b,-1) where b=sqrt(a^2+2)-sqrt(2)*a maintains a constant surface area 8, but the volume (8/3)*a*(sqrt(a^2+2) - sqrt(2)*a) varies from 0 to (8/3)*(sqrt(2)-1) = 1.10457and back to 0 again as 'a' varies from 0 to sqrt(2). The maximum volume occurs when a = b = sqrt(sqrt(2)-1) = 0.64359
While correct as stated, this example misses the point of the problem --- all the face areas can remain individually constant (which the above example fails to achieve) while the volume varies! While correctly computed, this misses the point of the problem: all the face areas need to remain individually constant (which the above fails to achieve) while the volume varies! Inscribe a tetrahedron in a cuboid (as suggested earlier), with vertices [a,0,0], [0,b,0], [0,0,c], [a,b,c]; all faces are congruent, with (via Cayley-Menger) 16*area^2 = 4 (a^2 b^2 + b^2 c^2 + c^2 a^2), 288*volume^2 = 32 a^2 b^2 c^2. Now choose b = a, c^2 = (1 - a^4)/(2*a^2); then as a varies from 0 to 1, each face area = 1/2 is constant, but volume = sqrt((1-a^4) a^2/18), WFL