# Questions tagged [geometric-constructions]

The geometric-constructions tag has no usage guidance.

31
questions

**2**

votes

**0**answers

49 views

### The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

I asked this question over a year ago on Math.StackExchange but I didn't get an answer.
In his famous treatise On spirals, Archimedean used a spiral to square the circle and trisect an angle. There ...

**4**

votes

**0**answers

81 views

### Compass and straightedge construction of Poncelet polygons

Gauss–Wantzel theorem states that
A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a ...

**10**

votes

**1**answer

647 views

### How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...

**5**

votes

**0**answers

150 views

### Are there any neusis-hard/neusis-complete problems?

I have lately been enjoying Richeson's Tales of Impossibility (see MAA review), an accessible book on the famous problems of Euclidean geometry including angle trisection/cube doubling/heptagon ...

**25**

votes

**2**answers

1k views

### 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 ...

**4**

votes

**2**answers

348 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 ...

**10**

votes

**1**answer

891 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.

**0**

votes

**0**answers

60 views

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

The remark [1] (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 ...

**1**

vote

**1**answer

95 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?

**6**

votes

**1**answer

256 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 ...

**2**

votes

**0**answers

605 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....

**36**

votes

**1**answer

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 ...

**3**

votes

**1**answer

273 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 ...

**5**

votes

**1**answer

157 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 ...

**4**

votes

**1**answer

709 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?

**9**

votes

**0**answers

354 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

829 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 ...

**10**

votes

**0**answers

631 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 $8$ ...

**2**

votes

**3**answers

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 ...

**1**

vote

**1**answer

232 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 $a\in\mathbb{C}$.
...

**4**

votes

**2**answers

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 ...

**4**

votes

**4**answers

900 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. ...

**6**

votes

**2**answers

597 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 ...

**9**

votes

**2**answers

1k 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 ...

**9**

votes

**2**answers

2k 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.

**26**

votes

**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 ...

**14**

votes

**3**answers

2k 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....

**16**

votes

**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 ...

**7**

votes

**3**answers

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.

**3**

votes

**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 ...

**4**

votes

**1**answer

1k 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 / \mathbb{...