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I've read and watched some lectures on euclidean geometry - not so advanced but I've seen the focus on constructions. Two instruments are used, compass and straightedge, I had the following doubts:

  • Why are there only two tools? Why only the compass and straightedge?

  • Are there more of these tools in other kinds of geometries?

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Wikipedia mentions several additional tools which might lead to possibility of constructing other things. –  Martin Sleziak Sep 10 '13 at 9:45
@MartinSleziak Thanks, Martin. I got interested in the Origami constructions. –  Voyska Sep 10 '13 at 9:51

2 Answers 2

To answer the first part of the question, these are Euclid's five postulates (per TL Heath's translation)

1 To draw a straight line from any point to any point

2 To produce a finite straight line continuously in a straight line

3 To describe a circle with any centre and any distance [equivalent to radius]

4 That all right angles are equal to one another

5 [Parallel postulate] That if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

The first three postulates describe constructions with straight edge and compass, and no other constructions are postulated. This is why the straight edge and compass constructions are the ones used in Euclidean geometry.

There are other devices known - angle trisectors, for example. But the mathematical development of Geometry proved more fruitful in other directions than the development of new mechanical devices for drawing geometrical figures.

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The compass and straight-edge are the easiest tools to make and use. The set constructed using these tools correspond to what I call $\mathbb{G}$, the complex numbers derived by way of the set {0,1}, and the closure of addition, subtraction, multiplication, division, and the extraction of square-root. For this reason, the mathematics is also relatively simple.

Euclid's text uses these tools, and it is as much tradition that more recent mathematics follow this. The same tools are used extensively in spherical and hyperbolic geometry too, for exactly the reasons of tradition and simple mathematics.

Other tools were well known in ancient and modern times as well, such as instruments to create ellipses, trisect angles, etc, and useful theoroms thus derived.

There is nothing stopping one implementing other tools, such as angle-trisectors and angle-qunisection. Many useful things can be found with such tools. Morley's theorem states that if you trisect the angles of a triangle, the crossings make an equalateral triangle, and with a trisection-tool, one can construct all prime-number polygons where $p=2^a3^b+1$. Adding qunisection permits $p=2^a3^b5^c+1$ to the mix, such as 11 and 151. But the tool takes some use, and usually just the outcome is shown. Likewise, one can find $\sqrt[3]{2}$ with such tools.

Of all of the uniform figures in higher dimensions, the snub cube and the snub dodecahedron alone would actually require trisections in their construction, since cubic equations need to be sought.

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Would you happen to know the equations for the stereographic projection of the snub cube? Kindly see this MSE post. –  Tito Piezas III Dec 5 '13 at 23:04
I didn't understand the stereographic projection equations you linked to. I don't have any of these, least of all on the class 3 and 6 figures (snub cube, dodecahedron). Sorry. –  wendy.krieger Dec 6 '13 at 8:59
Wendy: In the link, there is also a link to mathworld that explains those equations, namely I was intrigued by the last line of your answer, "...since cubic equations need to be sought." Do you have the explicit cubic, or any equation at all relating to the snub cube? –  Tito Piezas III Dec 6 '13 at 23:24
@TitoPiezasIII One makes extensive use of my 'class theorems' to get an idea of what kind of equation might be sought. Here, there are three free edges, which suggests a 3 to 1 mapping of lattice points, and hence a cubic. Wolfram actually confirms that the snub cube uses cubics, and the s Dodeca uses sextics. –  wendy.krieger Dec 7 '13 at 8:40

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