The geometric-constructions tag has no usage guidance.

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### Antoine's Necklace and positive Hausdorff/Lebesgue measure

I have the following question:
The usual construction of the Antoine's Necklace produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$.
I would like to know whether one could adapt ...

**4**

votes

**1**answer

543 views

### Focus of parabola using only a ruler

It is an easy exercise that using ruler and compass one find the focus of a given parabola.
Can one do the same using only a ruler? -- if not, why?

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**0**answers

282 views

### How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...

**28**

votes

**2**answers

684 views

### Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge ...

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**0**answers

324 views

### 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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votes

**3**answers

346 views

### 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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vote

**1**answer

211 views

### 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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**2**answers

954 views

### 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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**4**answers

757 views

### 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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**2**answers

470 views

### 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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votes

**2**answers

799 views

### 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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**2**answers

1k views

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

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

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**6**answers

2k views

### 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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**3**answers

788 views

### 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

1k views

### 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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**3**answers

2k views

### 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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**5**answers

1k views

### 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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**1**answer

836 views

### 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 / ...