Questions on the construction of geometrical figures using a limited set of "tools".

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4
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2answers
662 views

Resource for learning straightedge and compass constructions

Does anyone know a good resource for learning about straightedge and compass constructions besides "The Elements?" I tutor geometry to middle-schoolers and high-schoolers and thought that including ...
3
votes
0answers
20 views

how do the conic sections add to the possibilities of geometric construction?

If we are limited by what we can construct with compass and straight edge, then what becomes possible by expanding our toolkit to include all conic sections? The tools would be based on already ...
0
votes
1answer
26 views

smallest set of curves for constructing any real number and angle

If we are limited by what we can construct with compass and straight edge, then what are the fundamental curves required for constructing any real number? In other words, what is the smallest ...
23
votes
9answers
7k views

What is the (mathematical) point of geometric constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
3
votes
1answer
81 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
3
votes
1answer
80 views

Topology proof question?

How to prove that $X_1 := \{(x, y, z) ∈ \mathbb{R}^3 : x^2 + y^2 = 1\} / (S^1 × \{0\} )$ is homeomorphic to the union $X_2$ of two tangent spheres minus two points? What I know: Let $C$ be ...
2
votes
3answers
305 views

Doubling the cube neusis

Can anyone explain me in simple maths the neusis construction at http://en.wikipedia.org/wiki/Doubling_the_cube#Using_a_marked_ruler? Why and how does it produce the $\root 3 \of 2$?
0
votes
1answer
48 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
1
vote
1answer
34 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
3
votes
2answers
180 views

Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
-3
votes
0answers
127 views

Triangles on A Unit Square [closed]

A finite set S of unit squares is chosen out of a large grid of unit squares. The squares of S are tiled with isosceles right triangles of hypotenuse 2 so that the triangles do not overlap each other, ...
2
votes
1answer
62 views

Constructing an example of a infinite set of triangles on the rational line whose union does not contain the interior of a rectangle.

As part of my Topology course, I saw a proof the following proposition (as a consequence of Baire's category theorem): Proposition: Define $\triangledown_{t,h}$ as the interior of an equilateral ...
3
votes
0answers
51 views

Algebraic number of degree four that cannot be constructed with ruler and compass

The real number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$, where \begin{equation*} ...
6
votes
2answers
170 views

Geometry construction problem

Given two circles $S_1$ and $S_2$, a line $l_1$, and a length $a$ that is less than the sum of the diameters of the circles, construct a line $l$, parallel to $l_1$, so that the sum of the chords that ...
2
votes
0answers
42 views

Is there any construction method that yields all algebraic numbers?

Compass and Straightedge = rationals and square roots Origami = rationals, square roots and cube roots (I think) How far can we get if we use other tools, like rulers, protractors, pieces of string, ...
0
votes
2answers
54 views

Circle with center point and tangential to lines

I have defined Points all points (3 blue, and one green). All points have the same distance to A point. Yellow lines are bisectors. I have equations of AB and ...
8
votes
4answers
408 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
1
vote
2answers
37 views

Construct a midpoint of two parallel lines with only straightedge

Say I have a large plank of wood that I'm trying to cut in half the long way, but I only have a straightedge (no compass). How can I mark the midpoint between the long edges? (As a note, this isn't a ...
1
vote
1answer
133 views

Ruler-and-Compass metric

Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance ...
1
vote
2answers
51 views

Geometric Construction Problem

Given three non-collinear points A, B, and C, construct three circles that are pairwise tangent at these points. Are there any cases where such circles do not exist? I am not sure how to start the ...
1
vote
0answers
41 views

How to calculate the edge sizes of a Goldberg polyhedron?

I want to build a paper model of a Goldberg polyhedron, a Icosahedral G(2,2). But i cant find the formula to calculate the 2 sizes needed for the edges. For a G(4,1) the sizes are here : ...
6
votes
2answers
936 views

History of Compass/Straight Edge Construction

I'm interested in learning the origin of compass/straight-edge constructions. In particular, I am interested in the historical interplay between Euclid's axioms for plane geometry, and ...
0
votes
2answers
25 views

Find a point C on an infinite line AB which, when connecting two other points M and N, would form congruent angles

On an infinite line $AB$, find a point $C$ such that the rays $CM$ and $CN$ connecting $C$ with two given points $M$ and $N$ situated on the same side of $AB$ would form congruent angles with the ...
0
votes
1answer
21 views

construction of the rectangle with the highest area

I have 2 times a square with side length 2, 2 times a square with side length 3, 1 times a square with side length 4 and 1 times a square with side length 5. I have to create the rectangle with the ...
0
votes
1answer
27 views

Find circle for two points, one with given angle.

I have point A and B. I also have a vector v. How can I mathematically find a circle whose tangent at point C has the same angle as v where point C is the same as B and the circle also contains point ...
2
votes
1answer
42 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
3
votes
1answer
46 views

Where to place a bridge over the highway?

I've got a problem to solve. I had 2 different ideas which didn't actually work for all cases. On the both sides of highway there are 2 houses K and L (as in the attached picture). Line, that passes ...
0
votes
2answers
37 views

Trisection of an angle with straightedge and a compass

Suppose there exists an angle Z such that cos Z = -11\16 Prove or disprove that such an angle can be trisected with a straightedge and a compass. Well, we know that an angle is constructible iff its ...
1
vote
1answer
50 views

Visualizing construction of a certain function

Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\ 1) $f$ fixes the poles $(0,0,\pm1)$.\ 2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\ 3) $F$ is ...
2
votes
0answers
42 views

Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
1
vote
2answers
75 views

How to make π degree angle?

Can we make π degree angle? π is a decimal and angles are divided into minutes and seconds, but, I think (I'm not sure), we can still divide 1 degree into decimal parts (we can divide 1 degree into ...
2
votes
1answer
57 views

Triangle Construction being known its perimeter, height and angle

"Construct a triangle $ABC$ knowing his perimeter, the angle $\widehat{A}$ and the height relative to $BC$, i.e., $h_a$." It really looks to be an easy one, but I wasn't able to do it... :( Any hint? ...
0
votes
1answer
51 views

Tangent Circumference Construction

"Construct a circumference that is tangent to a given circumference and tangent to a line $r$ through a point $A$ of this line." I've done the line perpendicular to $r$ through $A$, cause we know ...
2
votes
1answer
38 views

Construct a triangle given an angle and two medians

Construct, with ruler and compass, a triangle $ABC$ knowing the angle $\widehat{A}$ and $m_a$ and $m_b$, where $m_a$ and $m_b$ are the medians relative to the vertices $A$ and $B$, respectively.
2
votes
0answers
73 views

History of the neusis construction of cube roots?

A simple neusis (marked ruler) construction of $\sqrt[3]{2}$ is given in many places, for example wikipedia. My question is: what is the history of this construction? As far as I can determine, all ...
9
votes
3answers
14k views

Compass-and-straightedge construction of the square root of a given line?

Given A straight line of arbitrary length The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude ...
11
votes
3answers
1k views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given 2 lines, one of the length of the hypotenuse and the other with the length of the sum of the 2 legs, construct ...
2
votes
0answers
49 views

Smallest field containing $\mathbb{Q}$ and closed under square root

I'm following Isaacs' Algebra and I need to prove that the field $K$ of constructible numbers is the smallest subfield of $\mathbb{C}$ such that is closed under taking square root. I already know ...
0
votes
0answers
8 views

Maximization of an integral based on Vittali covering

Let $f \in L^1(Y)$ and $f\geq 0$. Where $Y = [-1,1]^N$, suppose $\{B(y_k,\epsilon_k)\}_k$ forms a Vitalli covering of $Y$, satisfying \begin{equation}\begin{aligned} &(i) B(y_k,\epsilon_k) ...
0
votes
1answer
34 views

Constructing a special point in quadrilateral

This problem appeared in my mind when I was working on another problem, I found it interesting but still don't know how to solve it yet. So i decided to post it here, I hope we can discuss and you guy ...
5
votes
1answer
123 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
0
votes
1answer
29 views

Geometric Construction

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and ...
3
votes
1answer
65 views

Regular Polyspiral (Geometry)

Criteria: -Each side of every polygon has to be the same length. -Every successive polygon has to have one more side. -Each additional polygon has to start on the opposite right side (assuming the ...
6
votes
4answers
209 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. ...
2
votes
0answers
42 views

Is this element constructible from this elements?

Let the figure below. According to same notation of the figure verify if it's possible to construct the point $\displaystyle \zeta=e^{\frac{2\pi i}{13}}$ with straight-edge and compass from ...
1
vote
1answer
62 views

Dynamic Geometry Software for teach construction

I would like to know about a software that will help me show construction steps to the students with using a Compass/Straight Edge/Protractor/Divider. Here is example video below. ...
6
votes
1answer
67 views

Are there numbers that we can't get with a usual compass and ruler, but can get with 3D compass and ruler?

If we have unit segment, we can use a compass and ruler to make segments whose length represents many numbers (all rational, sqrt(2)), but there are "unreachable" ...
8
votes
4answers
202 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
2
votes
2answers
266 views

Construction of a triangle, given: side, sum of the other sides and angle between them.

Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.
10
votes
0answers
161 views

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean ...