The Beautiful Turns
Imagine for yourself the figure we know as the baraed (equilateral triangle). It possesses three equal lines, each turning (angling) by an equal amount to form a closed figure. It may be oriented in any of six positions, in which, were it to be cleaved in two, it would form two shapes, identical but inverse to each other. These six positions have two different visually distinguishable orientations, one of which is wide at the top, the other of which is wide at the bottom.
These two shapes, the high baraed and the low baraed, can be superimposed upon each other. If their sides are placed upon each other, they merge to form a feraen (diamond). Move them together so that their centres overlap, we are presented with the image of a soarues (six-pointed star). Each side is half of the length of those in the original baraeda, and the turns (the word ‘turns’ is translated from Faerouhaiaouan ‘holaia’, and the concept is similar to that of ‘angles’ between the lines alternate between their original size, and twice their original size. If one turn of the original size and one turn of the doubled size are isolated, and placed embracing each other, a new, perfectly straight line is formed. One quickly learns that six turns of the original size, or three turns of the double size, can be placed together to form a complete rotation.
The original turns are the components of a baraed. However, the doubled turns, when reconstructed into a figure, form a haraed (hexagon). One can continue to slightly increase the size of each of these turns, increasing the number of equal lines in a figure. However, this can only be continued until the size of the turn has reached that of three turns of a baraed. At this point, the line becomes straight. However, infinitely close to this point, exists the hai (circle)- the most perfect and singular of all figures, composed of an infinite number of equal segments, each separated by a turn infinitely close to that of the combination of three turns of a baraed.
Consider that the turns of six baraeda, placed together, form a closed figure. These six baraeda are twice as many as the three bareda required to form a straight line. Thus, we understand that while three baraeda turns are enough to create a linear form, six baraeda turns are enough to rotate a form entirely against itself.
The shape created by these six triangles is the haraed. Once again, we see that the combination of two baraeda turns results in the creation of a haraed turn. Six haraeda turns, each twice as large as a single baraed turn, are needed to close the figure. This pattern can be observed to continue itself as one enlarges the turn. Each widening results in the need for more lines to close a figure, until the infinite point is reached, at which the hai is formed.
Imagine for yourself the figure we know as the maed (square). It possesses four equal lines, each turning around by an equal amount to form a closed figure. It may be oriented in any of eight positions, in which, were it to be cleaved in two, it would form two shapes, identical but inverse to each other. These six positions have two different visually distinguishable orientations, one of which is equally wide at the top and bottom, the other of which is wide at the centre and thin at the top and bottom.
When the turns of two maeda are placed together, they instantly form a straight line. Likewise, the combination of four turns results in a complete rotation. Knowing that three and six baraeda turns were required to perform the same tasks, it is demonstrated that the turn of the maed is equal to the turn of half of three turns of a baraed.
The straight line can be defined only by the angles of these two figures. It remains the closest knowable value to that of the turn of a hai- a beautiful and sacred value that can never become more than a crude approximation...
The preceding is a translated extract from ‘Holaia Haiaoua’ (The Beautiful Turns), the magnum opus of the Faerouhaiaouan Mathematician and Philosopher Raedae Surahaila. Much was to come of his groundbreaking inquiries into geometry, fractions, and the infinite.
Imagine for yourself the figure we know as the baraed (equilateral triangle). It possesses three equal lines, each turning (angling) by an equal amount to form a closed figure. It may be oriented in any of six positions, in which, were it to be cleaved in two, it would form two shapes, identical but inverse to each other. These six positions have two different visually distinguishable orientations, one of which is wide at the top, the other of which is wide at the bottom.
These two shapes, the high baraed and the low baraed, can be superimposed upon each other. If their sides are placed upon each other, they merge to form a feraen (diamond). Move them together so that their centres overlap, we are presented with the image of a soarues (six-pointed star). Each side is half of the length of those in the original baraeda, and the turns (the word ‘turns’ is translated from Faerouhaiaouan ‘holaia’, and the concept is similar to that of ‘angles’ between the lines alternate between their original size, and twice their original size. If one turn of the original size and one turn of the doubled size are isolated, and placed embracing each other, a new, perfectly straight line is formed. One quickly learns that six turns of the original size, or three turns of the double size, can be placed together to form a complete rotation.
The original turns are the components of a baraed. However, the doubled turns, when reconstructed into a figure, form a haraed (hexagon). One can continue to slightly increase the size of each of these turns, increasing the number of equal lines in a figure. However, this can only be continued until the size of the turn has reached that of three turns of a baraed. At this point, the line becomes straight. However, infinitely close to this point, exists the hai (circle)- the most perfect and singular of all figures, composed of an infinite number of equal segments, each separated by a turn infinitely close to that of the combination of three turns of a baraed.
Consider that the turns of six baraeda, placed together, form a closed figure. These six baraeda are twice as many as the three bareda required to form a straight line. Thus, we understand that while three baraeda turns are enough to create a linear form, six baraeda turns are enough to rotate a form entirely against itself.
The shape created by these six triangles is the haraed. Once again, we see that the combination of two baraeda turns results in the creation of a haraed turn. Six haraeda turns, each twice as large as a single baraed turn, are needed to close the figure. This pattern can be observed to continue itself as one enlarges the turn. Each widening results in the need for more lines to close a figure, until the infinite point is reached, at which the hai is formed.
Imagine for yourself the figure we know as the maed (square). It possesses four equal lines, each turning around by an equal amount to form a closed figure. It may be oriented in any of eight positions, in which, were it to be cleaved in two, it would form two shapes, identical but inverse to each other. These six positions have two different visually distinguishable orientations, one of which is equally wide at the top and bottom, the other of which is wide at the centre and thin at the top and bottom.
When the turns of two maeda are placed together, they instantly form a straight line. Likewise, the combination of four turns results in a complete rotation. Knowing that three and six baraeda turns were required to perform the same tasks, it is demonstrated that the turn of the maed is equal to the turn of half of three turns of a baraed.
The straight line can be defined only by the angles of these two figures. It remains the closest knowable value to that of the turn of a hai- a beautiful and sacred value that can never become more than a crude approximation...
The preceding is a translated extract from ‘Holaia Haiaoua’ (The Beautiful Turns), the magnum opus of the Faerouhaiaouan Mathematician and Philosopher Raedae Surahaila. Much was to come of his groundbreaking inquiries into geometry, fractions, and the infinite.