2¹⁰ &asymp 1000, so
2²⁰ &asymp 1000000, a million, 10⁶;
2³⁰ &asymp 1000000000, a billion, 10⁹;
2⁴⁰ &asymp 1000000000000, a trillion, 10¹²
and then 2⁸⁰ &asymp a trillion trillion, 10²⁴.

Page 75:

81 is 3⁴, so 80 &asymp 81 implies that
3¹² &asymp 80³
Now use the approximation 3¹² &asymp 2¹⁹ to give
2¹⁹ &asymp 80³ = 5³ × 2¹².
Dividing by 2⁹,
2¹⁰ &asymp 5³ × 2³ = 10³, i.e. 1024 &asymp 1000.

A perfect harmonic seventh is given by the ratio 7/4. A perfect harmonic fourth is given by the ratio 4/3. Two fourths give 16/9, and the approximation 16/9 &asymp 7/4 is equivalent to 64 &asymp 63.

Hexadecimal FF means 255 in decimal notation, as already explained. Then 69 in hexadecimal means six sixteens plus nine, which is 105 in decimal notation. Hexadecimal B4 means eleven sixteens plus four, which is 180.

The number of different colours is 256 × 256 × 256 = (2⁸)³ = 2²⁴ &asymp 16 million.

It may come as a surprising fact that when a cube is suspended from one vertex, and then bisected by a horizontal plane, the cut makes a regular hexagon.

This graphic showing the hexagon was created by John Augar.

This shows the plane of colours with R + G + B = 384, forming the central hexagon.

Three dimensions for the location of the camera. Two for the direction in which the lens points, one more for the angle of the shot, totalling six. If you limit yourself to pictures which respect the vertical direction, however, only five. For the spotlight, another three dimensions for location and two for the direction of the shot. If it has a shade with some asymmetry, then there is another dimension.