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Sphere

This is ok:
The area and volume of a sphere are 1/2 that of the box it came in.

## Landmarks

 volume of a sphere - "1/2 that of the box it came in" (-5%, ±<1%)
 1 exact volume of a box of size 1 0.5236 ±<<1% volume of a sphere of diameter 1 0.52 ±<1% 0.5 -5%, ±<1%

 area of a sphere - "1/2 that of the box it came in" (-5%, ±<1%)
 6 exact area of a box of size 1 3.142 ±<<1% area of a sphere of diameter 1 3 -5%, ±<1%

 "pi" - 3 (-5%, ±<1%) 3.1 (-1%, ±<1%)
 3.1416 ±<<1% 3.14 ±<<1% (~-0.05%) 3.1 -1%, ±<1% 3 -4.5%, ±<0.01% 3 -5%, ±<1% 1 -68%, ±<1% 1 -70%, ±<2%

## Notes

```Currently in pre-first-draft state...
Do `vs cube of same mumble' landmarks.
Need distinct `simple closed volume' page?
Growth relations.
4.2, 4.19 concept.
```

Sphere vs cube
constantside&diam.areavolumerelation
size mcube 1 m 6 m2 1 m3 1.91 × sphere of same size
sphere 1 m 3.142 m2 0.5236 m3 0.524 × cube of same size
area m2cube 0.4082 m 1 m2 0.06804 m3 0.724 × sphere of same area
sphere 0.5642 m 1 m2 0.09403m3 1.38 × cube of same area
volume m3cube 1 m 6 m2 1 m3 1.24 × sphere of same volume
sphere 1.241 m 4.836 m2 1 m3 0.806 × cube of same volume

```Sphere vs cube of same size (0.5, -5%)
Sphere vs cube of same area
4/3 | 3/4 (4%)  1 1/4 | 0.8 (10%)  1.3 (6%)  "+40%" | 5/7 (2%)  1/2 (50%)
Sphere vs cube of same volume
4/3 | 3/4 (8%)  1 1/4 | 0.8 (1%)   1.3 (5%)
c v/a  4/3 <10%
c vol  "25%" (1%)
c area "40%" (2%)
deformation image to keep the two straight
```

Sphere diameter shrinking vs area required to enclose 1 m3.
diamareavolN spherestotal area
1 m3.142 m20.52361.9106
10-1 m3.142 × 10-2 m25.236e-041.910 × 1036 × 101
10-2 m3.142 × 10-4 m25.236e-071.910 × 1066 × 102
10-3 m3.142 × 10-6 m25.236e-101.910 × 1096 × 103
10-4 m3.142 × 10-8 m25.236e-131.910 × 10126 × 104
10-5 m3.142 × 10-10 m25.236e-161.910 × 10156 × 105
10-6 m3.142 × 10-12 m25.236e-191.910 × 10186 × 106
Note volume increasing as r3 (10-3 per 10-1), but area only as r2 (10-2 per 10-1).
[...Simplify...]

Size of shrinking spheres vs area for 1 m3 worth.
diameterareavolN spherestotal area
× 1/2 (m) × 1/4 (m2) × 1/8 (m3) × 8 × 2 (m2)
(1/1) 1.241 m4.838 m21.000 m314.834 m2
(1/2) 0.6205 m1.210 m21.251 × 10-1 m389.670 m2
(1/4) 0.3103 m0.3024 m21.564 × 10-2 m36419.34 m2
(1/8) 0.1551 m0.7560 × 10-1 m21.955 × 10-3 m351238.68 m2
(1/16) 0.7756 × 10-1 m1.890 × 10-2 m22.443 × 10-4 m3409377.36 m2
(1/32) 0.3878 × 10-1 m0.4725 × 10-2 m23.054 × 10-5 m332744154.7 m2
(1/64) 1.939 × 10-2 m1.1812 × 10-3 m20.3817 × 10-5 m3261954309.4 m2
(1/128) 0.9695 × 10-2 m2.953 × 10-4 m20.4772 × 10-6 m32095632618.9 m2
(1/256) 0.4848 × 10-2 m0.7383 × 10-4 m20.5965 × 10-7 m3167650521238 m2
(1/512) 2.424 × 10-3 m1.8457 × 10-5 m20.7456 × 10-8 m31341204192475 m2
(1/1024) 1.212 × 10-3 m0.4614 × 10-5 m20.9320 × 10-9-10 m310729633504951 m2
(1/2048) 0.6060 × 10-3 m1.154 × 10-6 m21.165 × 10-10 m385837067979901 m2

 A View from the Back of the Envelope Comments encouraged. - Mitchell N Charity

```Doables:
In need of overhaul.
```History: