# Questions tagged [geometric-constructions]

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### Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?

I am hoping that the brilliant MathOverflow geometers can help me out. Question 1. Suppose that I have a fixed finite-length straightedge and fixed finite-size compass. Can I still construct all ...
273 views

### Domino tiling obtained from space-filling curves, is possible to predict basic properties?

Periodic and aperiodic domino tiling systems can be obtained by the following construction rules: Draw a regular square grid n×n of n2 cells. Select a space-filling curve that is consistent with ...
864 views

### How can we find n points on a plane so that as many pairs of points as possible have the same distance?

There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
52 views

### Examples of geometrical interpretations for sequences of particular values of Dirichlet series

The remark  (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
93 views

### Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
148 views

### Algorithm to decide whether two constructible numbers are equal?

The set of constructible numbers https://en.wikipedia.org/wiki/Constructible_number is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am ...
211 views

### What makes a geometric construction more or less stable?

I'm not entirely sure if this is "research level" math or not, but I asked on Math.SE and it was suggested I try asking here, so hopefully it's of interest to this community. (Original question on M....
1k views

### Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid)

Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction ...
271 views

### Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
151 views

### 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 ...
699 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?
333 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 ...
811 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 ...
616 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 $8$ ...
2k 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 ...
228 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 $a\in\mathbb{C}$. ...
1k 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 ...
870 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. ...
577 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 ...
992 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 ...
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.
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 ...
1k 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 http://en.wikipedia....
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 ...
3k 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.
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 / \mathbb{...