The Importance of Irregular Triangles

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Now FINALLY we can go back to the original question. How was Bill Cipher born? It has to do with the law of nature isosceles triangles have.

Remember; Every shape in Flatland (starting with equilateral triangles and every shape above), has a law of nature that if they have a son, that son will have one more side than his father. However, this law of nature does NOT apply to any isosceles triangles or irregular shapes.
"This rule applies not always to the Tradesmen, and still less often to the Soldiers, and to the Workmen; who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal. With them, the Law of Nature does not hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles still."
Now you're probably even more confused about the specifics of Bill's birth. Not to worry, I've got an explanation. You see, isosceles also have this other law of nature.
"Nevertheless, all hope is not shut out, even from the Isosceles, that his posterity may ultimately rise above his degraded condition. ... It is with us a Law of Nature that the brain of the Isosceles class shall begin at half a degree, or thirty minutes, and shall increase (if it increases at all) by half a degree in every generation; until the goal of 60° is reached, when the condition of serfdom is quitted, and the freeman enters the class of Regulars."
In summary, isosceles can give birth to a son whose angles become closer to an equal-sided triangle with each generation instead of an extra side. These kinds of births must be carefully planned out and considered uncommon. Here's how;
"Intermarriages (arranged by the Priests) between the sons and daughters of these more intellectual members of the lower classes generally result in an offspring approximating still more to the type of the Equal-Sided Triangle."
It's all a bit uncanny, don't you think? The large population of Isosceles has one goal, and it's to make the perfect child. The triangles with angles closer to the perfect 60 degrees and daughters of triangles with their angles also being less irregular are paired up in arranged marriages. Then they'll try and have many kids in the hope they have a son whose angles are closer to that perfect 60 degrees. Then the cycle continues for generations, each offspring possibly becoming closer and closer to perfection.

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- Bill is from the 2nd dimension, a world filled with different kinds of shapes and lines known as Flatland.
- Bill as an equilateral triangle is part of one of the lowest classes in society and is thus mistreated. Flatland has a social hierarchy that has triangles at the bottom, squares next, then pentagons, etc. Circles are at the top of society. At the very bottom (not considered part of society) are irregular triangles and irregular shapes, and the women who are straight lines.
- Unlike equal-sided shapes (every child has one more side than his father), Irregular triangles over many generations of careful breeding can have equilateral triangle sons.

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