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Originally shared by Richard GreenTrisecting the angle... using origami
A problem that the ancient Greeks considered is whether it is possible to trisect an arbitrary angle using a straightedge and compass. The answer is now known to be no, but remarkably, it is possible to cut an arbitrary acute angle into three equal parts by using origami. The main picture shows how, but what exactly does this mean?
Straightedge and compass
The straightedge and compass are mathematically idealized drawing tools. The compass can be opened arbitrarily wide, but has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle. The straightedge is infinitely long, but it has no markings on it and has only one edge, unlike an ordinary ruler. It can only be used to draw a line segment between two points or to extend an existing line. The problem then becomes to find exact constructions for given lengths with reference to a unit length, using only these two instruments. Although this sounds very artificial, it can be important in drafting, and in the science of weights and measures.
The Greeks were unable to find a straightedge and compass algorithm to divide an arbitrary angle into three equal pieces, and with good cause: for most angles, this is impossible. Note that this does not mean that nobody has been clever enough to think up a method to solve the problem; it means that it is not possible, within the constraints of the problem, to trisect an arbitrary angle perfectly, even though there may be excellent approximate solutions. The standard proof of this result uses relatively modern ideas from abstract algebra.
The axioms of origami
It is also possible to pin down some axioms for origami, the traditional Japanese art of paper folding. The second picture shows seven such axioms (numbered O1-O7) which are known as the Huzita-Hatori axioms (or sometimes the Huzita-Justin axioms). Although it is possible to find a shorter set of axioms, these seven have the advantage of including the complete set of possible single origami folds. Axioms O1 to O4, taken together, have weaker constructive powers than the straightedge and compass framework. The first five axioms, O1 to O5, taken together, are equal in power to the straightedge and compass.
Axiom O6 involves a kind of sliding move called neusis which is illegal in the straightedge and compass framework. Adding axiom O6 to the first five gives a more powerful system than straightedge and compass. An example of this power is the construction in the first picture: the origami axioms are capable of trisecting an arbitrary angle perfectly, whereas the straightedge and compass construction is not.
The diagrams come from the book Origami and Geometric Constructions by Robert J. Lang, available on his website, http://www.langorigami.com. The website also contains many elaborate works of origami art by the author.
#mathematics #origami
A problem that the ancient Greeks considered is whether it is possible to trisect an arbitrary angle using a straightedge and compass. The answer is now known to be no, but remarkably, it is possible to cut an arbitrary acute angle into three equal parts by using origami. The main picture shows how, but what exactly does this mean?
Straightedge and compass
The straightedge and compass are mathematically idealized drawing tools. The compass can be opened arbitrarily wide, but has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle. The straightedge is infinitely long, but it has no markings on it and has only one edge, unlike an ordinary ruler. It can only be used to draw a line segment between two points or to extend an existing line. The problem then becomes to find exact constructions for given lengths with reference to a unit length, using only these two instruments. Although this sounds very artificial, it can be important in drafting, and in the science of weights and measures.
The Greeks were unable to find a straightedge and compass algorithm to divide an arbitrary angle into three equal pieces, and with good cause: for most angles, this is impossible. Note that this does not mean that nobody has been clever enough to think up a method to solve the problem; it means that it is not possible, within the constraints of the problem, to trisect an arbitrary angle perfectly, even though there may be excellent approximate solutions. The standard proof of this result uses relatively modern ideas from abstract algebra.
The axioms of origami
It is also possible to pin down some axioms for origami, the traditional Japanese art of paper folding. The second picture shows seven such axioms (numbered O1-O7) which are known as the Huzita-Hatori axioms (or sometimes the Huzita-Justin axioms). Although it is possible to find a shorter set of axioms, these seven have the advantage of including the complete set of possible single origami folds. Axioms O1 to O4, taken together, have weaker constructive powers than the straightedge and compass framework. The first five axioms, O1 to O5, taken together, are equal in power to the straightedge and compass.
Axiom O6 involves a kind of sliding move called neusis which is illegal in the straightedge and compass framework. Adding axiom O6 to the first five gives a more powerful system than straightedge and compass. An example of this power is the construction in the first picture: the origami axioms are capable of trisecting an arbitrary angle perfectly, whereas the straightedge and compass construction is not.
The diagrams come from the book Origami and Geometric Constructions by Robert J. Lang, available on his website, http://www.langorigami.com. The website also contains many elaborate works of origami art by the author.
#mathematics #origami
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