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### The construction of the 257gon

If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...

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### Mid point with set square?

Is it possible to construct the midpoint of a segment in the hyperbolic plane
using the set square only?
With the set square one can
draw the line through the given two points and
drop the ...

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### Algebraic characterization of points constructible by compass and straightedge

The typical characterization of points constructible by compass and straightedge is the following:
Let $S\subseteq\mathbb{C}$ with $0,1\in S$, $K_0 = \mathbb{Q}(S\cup \bar{S})$ and $a\in\mathbb{C}$.
...

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### How to draw Archimedean-Galileo spiral?

It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and ...

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### What are the lengths that can be constructed with straightedge but without compass?

Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. ...

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### Bolyai's construction

Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question.
It is a Compass-and-straightedge construction of asymptotically parallel line in ...

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### Is there a ruler and compass construction of the common perpendicular of two geodesics in H^3?

Assume we have two geodesics in the Poincaré ball model of $\mathbb{H}^3$, viewed as arcs intersecting the boundary of and contained in the Euclidean unit sphere in $\mathbb{R}^3$. Is there a ruler ...

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### Is there a compass and straight edge construction of parallel lines in hyperbolic geometry?

Is there a compass and straight edge construction of parallel lines in hyperbolic geometry?
That is given a line and a point not on the line construct one of the line parallel to the given line.

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### How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions ...

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### Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and ...

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**1**answer

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### Origami Constructions: Intersecting two Circles

It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry ...

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### how to construct a spherical dodecahedron?

using only a spherical ruler (to construct great lines) and a pair of compasses, how can you construct a regular dodecahedron on the surface of the sphere? thank you very much.

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### Approximate solutions for trisecting the angle or squaring the circle

Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a ...

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### On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / ...