Wednesday, September 29, 2004

Rubber Geometry

And I don't mean topology...

O = center of circle; V is the top of the circle, M is a point on OV, and E is a point on the circle at the same level as M

I recently happened across the start of a very detailed commentary on the Evangelion TV series, which brought home to me that not only is the anime physics of necessity rubber, but so is some of the geometry.

The diagram shows a sketch of the NERV geofront, a sphere notionally 89% full according to ep 20 i.e. 11% of it is open as a spherical cap. From the latter page we see that the volume of a spherical cap of height h (VM in the diagram) cut from a sphere radius R (OE or OV) is πh²(3R-h)/3; the volume of the whole sphere is 4πR³/3. If we let x = h/R, then volume of cap/volume of sphere is x²(3-x)/4, = 0.11 in our case.

A little computation shows that x ~ 0.412, so OM = 0.588R, and since OME is a right-angled triangle, (ME/R)² = 1 - (0.588)², so ME is 0.809R.

But it is also claimed that

The Geofront is a sphere with a diameter of 13.75 km). However, 89% of it is buried and only the top part (900 m high and 6 km across) is open.

i.e. R = 6.875 km, 2ME = 6km and h = 0.9km. The drawing is done with R and h in those proportions. There is no way to reconcile these figures, which can be derived with a little high-school geometry.

Taking the R and h values, we have ME² = 6.875² - 5.975² = 11.565, so ME = 3.4km, or about 7km across, which is not too bad, but x = h/R = 0.131, giving a fraction of volume in the cap of about 1.2% of the sphere, about 1/10 of what was stated.

So perhaps it might have been a slipped decimal point, 1.1% recorded as 11%, and the rest is history.

Later -- Of course the 89% figure might be right, if NERV HQ is like an iceberg; nine-tenths hidden. And there are a lot of vast caverns shown in the depths below that might well use up a lot of space.

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