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I want to ask about the principle or the idea that people who invented the geometric method of drawing a regular pentagon depended on. I know how to follow the steps described below to draw the pentagon but I want to know how these steps led us to a regular pentagon. They must have depended on a rule, law, idea or something so what is it?

Note: I am talking about the geometric method explained in the pictures not the divider method.

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There are many ways to do this, but they all depend on the same basic fact that the sines and cosines of the angles $36°$ and $72°$ involve $\sqrt 5$. For example $\cos 36° = (1 + \sqrt 5)/4$. Since $\sqrt 5 = \sqrt{1^2 + 2^2}$, These angles can be constructed by starting from a right-angled triangle with sides $1$, $2$, and hypotenuse $\sqrt 5$.

In the OP's construction, is should be obvious how the $\sqrt 5$ appears in the length of AE and AB.

Euclid gave a (rather long winded) construction in Elements, book IV, proposition 11 and proved it was correct using only geometry.

It is usually easier to prove that "quicker" constructions like the OP's are correct using trig, rather than pure geometry.

The more general question of "which regular polygons can be constructed using only ruler and compasses" (i.e. no measuring instruments allowed) was answered by Gauss. Only 31 such polygons with an odd number of sides are known, the smallest ones having 3, 5, 15, 17, and 51 sides. The largest known one has 4,294,967,295 sides. It is an unsolved mathematical problem whether any more exist, and if so whether there are a finite or infinite number of them.

See:

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An number of methods have been devised to construct pentagons, among them Richmond's method and Carlyle circles. Classical methods are based on mathematics: trigonometry and Pythagoras's theory of triangles.

An animation of the method you are asking about is in the Wikipedia article of Pentagons.

pentagon animation
Click image to expand and see animation

Wolfram, also gives a good description of the method,

Euclid showed how to inscribe a regular pentagon in a circle. Ptolemy also gave a ruler and compass construction for the pentagon in his epoch-making work The Almagest.

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  • $\begingroup$ Giving references to outside links is not considered a good way to answer on the SE network. You should include some explanation here itself. $\endgroup$ – shivams Sep 27 '15 at 9:21
  • $\begingroup$ @shivams - Yes and no. Link only answers are not acceptable within the SE network. That said, I wouldn't call this a link only answer. This answer provides a bit more explanation along with the two links and a relevant quote from one of the links. I have edited the animation into answer. $\endgroup$ – user16 Sep 27 '15 at 16:12

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