- Circles in the sky, measured in units of time and calendar.
- Real circles, measured by their diameters, the value \( \pi\) measures the roll against diameter.
- Arcs, which are measured by right variation at a distance.

Circles in the sky are measured according to the day-cycle (equatorial circles give a zenith-marker pointing to siderial noon), and the elliptic traces out the movements of the sun, the degree corresponding close to a day. In either case, these divisions are properly dealt with with the derivation of time units.

Radian measure is usually applied to small variations over a distance, such as gradients (1:12 or 8.3% etc), or to distances at a distance (the military 'mil' is taken to be yards at 1000 yards, and \(\pi=3.2\) in NATO countries, or \(\pi=3.15\) in Sweden. Because this system closely matches the units inferred in the Taylor series, and the radian is the angle coherent in the relation of torque = moment of force, it is used in defining systems. However, it is not essential for geometry, and I use it mainly when i write programs.

The real-circle is one which one might visit all parts of, or hold in the hand. It is given by its

*diameter*, symbol Ø. In Sir Thomas Heath's "history of greek mathematics", the sumerian angle for real circle was to divide the circumference into 180 ells, of 24 digits, the diameter taken to be 60 ells. This is the unit that i divide into 120 parts, and so forth.

The measures of solid angle are by area, there are three units, the steradian, the degree excess, and the square degree.

1s00 (sphere=1)

System | Plane Angle | Solid Angle |

Natural | \( 2\pi\) radians | \(4\pi\) Steradians |

Common | \(360\) degrees | \(720\) degrees Excess |

Astronomy | \(360\) degrees | \(129600/\pi\) square degrees |

Twelfty | 1c00 (circle = 1) | 1s00 (sphere = 1) |

The units of the twelfty system relate to the fraction of whole space occupied by a thing. The letter c or s or g stand for the complete space in 2, 3, 4 dimensions, and replaces the radix-point. The solid angle of a cube is s15 at the vertex (ie 1/8), but s30 on the edge (1/4), The solid angle at the edge is equal to the dihedral or margin angle, which is c30. The largest solid angle by any vertex of a regular solid belongs to the twelftychoron, {5,3,3} which gives g38.24.

## Twelfty angle and astronomy

When one supposes a division of the circle to 12 signs of 10 degrees, and thinks consistantly of this, it becomes easier to divide the month and year into 120 parts, especially as these are in effect ¼ and 3 days respectively. So one can guage the phase of the moon, and its position of sky, and even with these crude measures of eye-estimate, derive a time to the nearest hour. Full moon is taken as c60, new moon as c00. So if you see the moon is waxing, and its at c40, and at c50 in the sky, (zenith is 60, nadir is 0) then the sun is trailing it by 40, makes it c10, or 2am.

With the siderial time, one would convert the year to a fraction. It's 31/8/19 today, but since my calendar begins on 1/3, we have 6 full months and no days. So it's y60. The present time of 21:13, translates to twelfty as 9.73, or d96.1. So the current siderial time is 96.1 - 60 = 36.1 (or 7 am).

It is possible to use a common angle through-out.

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