Happiness! I'm here to provide a brief lesson in geometry. Specifically, we'll be discussing circles.
A circle can be described by the longest distance across it. This distance is called the circle's cross-length, or CL for short. Cross-lengths are of course measured in units of length: blades, reaches, throws, or even walks.
The other length associated with a circle is the distance around the outside of the circle. This is called its around-length. The ratio between a circle's around-length and its cross-length is a constant that is called "Around-Number", which we'll abbreviate as AN. In base 7, Around-Number is approximately 3.0663652, but for many uses 3.1 is close enough.
The space inside a circle is called the circle's ground. Consider a circle with a cross-length of 1 reach. This circle has a ground of 1 circled reach.
What about a circle with a cross-length of 2 reaches? This circle, it can be shown, has a ground equivalent to 4 times the ground of the previous circle, or 4 circled reaches. As a rule, a circle with a cross-length of CL units has a ground of CL multiplied by CL circled units. We thus call multiplication of a number by itself "circling": 2 circled is 4, or 2° = 4. The inverse of circling is known as "crossing": 4 crossed is 2, or 4^{Φ} = 2.
A ring is a circle with another circle removed from its center. The ground of a ring is the ground of the larger circle (that is, its cross-length circled) minus the ground of the smaller circle: G_{R} = CL_{L}° - CL_{S}°
If a circle or ring is divided radially into a number of equal sectors, naturally the ground of each sector is the ground of the circle or ring divided by that number. That number is called the share-number of the sector. As a general equation, we can say this:
G_{S} = (CL_{L}° - CL_{S}°) / SN
A sector has a thickness of CL_{L} - CL_{S} and inner and outer boundaries of CL_{S} * AN / SN and CL_{L} * AN / SN respectively.
A common geometrical task is to divide a circle into many equal sectors. For instance, this is useful in apportioning a piece of land such as in a farm, a quarry, or a city. There are many ways to accomplish this task. A simple way is to create boundary circles of steadily increasing cross-lengths, and divide each ring into a number of sectors equal to its ground divided by a standard sector size.
Here is one such division. The central circle has a cross-length of 2 reaches and is divided into 4 sectors which each have a ground of 1 circled reach. Each successive circle has a cross-length that is 2 reaches larger and is divided into 4 times X sectors, where X also increases by 2. This division is easy to calculate and lay out, but the staggered pattern of the sectors may not be ideal for some applications.
In some situations, such as in laying out a city, it's desirable to keep the number of divisions constant, or at least multiples of each other, as the cross-length increases, so as to keep the sectors aligned with each other. This means that the circles will get closer together. If the sector thickness becomes too small, doubling the share-number makes it almost twice as thick. To calculate the cross-length of each new circle, we solve for the cross-length of the larger circle:
CL_{L}° = CL_{S}° + (SN * G_{S}) CL_{L} = (CL_{S}° + (SN * G_{S}))^{Φ} |
Here is an example division, with a central lot of the same size as the others, starting with a share-number of 6, and a minimum sector thickness of half the cross-length of the central lot. This division is commonly used for cities, as the 7 central lots are aesthetically pleasing and outside of them there are streets in all 4 cardinal directions. A circled throw is a common lot size.
As sectors get farther away from the center, their borders become less curved, becoming more similar to a rectangle. The ground of a rectangle can be determined in this way. Consider a near-rectangular sector of thickness T and an inner boundary of B. The small circle of the sector has a cross-length of CL where CL is very large. SN in this case is (AN * CL) / B. The large circle has a cross-length of CL + (2 *T).
G_{r} = ((CL + 2 * T)° - CL°) / ((AN * CL) / B) G_{r} = (CL° + 4 * CL * T + 4 * T° - CL°) * B / (AN * CL) G_{r} = (4 * CL * T + 4 * T°) * B / (AN * CL) G_{r} = (4 * CL * T * B / (AN * CL) ) + (4 * T° * B / (AN * CL)) G_{r} = (4 * T * B / AN) + (4 * T° * B / (AN * CL)) |
Since CL is very large compared to T°, the second argument to the addition is negligible.
G_{r} = 4 * T * B / AN
That concludes the lesson. I hope that it has been educational. Happiness until we next meet.
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