Euclid-style constructions

[lpetrich]

Euclid's Elements I've seen called the most successful mathematics textbook of all time. Written around 300 BCE, it has been widely copied and often translated over the centuries. Mathematicians in recent centuries have pointed out some flaws in it, and its style of construction is not very useful for many practical applications or much of the rest of mathematics, but it is a historical monument.

•  Euclid's Elements
• Euclid's Elements, Euclid - an English translation

Euclid-style constructions as a puzzle game:

• Euclidea - Geometric Constructions Game with Straightedge and Compass
• Euclid: The Game
• Euclid: The Game - Tutorial

See how few steps one can use in constructing a proof.

In Euclid-style constructions, one uses a straightedge and compass to make lines and circles.

An extension of it is the  Neusis construction, with a marked straightedge, but it follows in the tradition.

As to not being very useful, many applications of geometry are based on analytic geometry, with its use of coordinate systems, and such forms of mathematics as the calculus are associated with analytic geometry.

 

[lpetrich]

Euclid found that he could not prove everything, and his axioms were his starting points. They are divided into two sets of five.

Common Notions:

1. Things which equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things which coincide with one another equal one another.
5. The whole is greater than the part.

Postulates:

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and radius.
4. That all right angles equal one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Most of them look like common sense, with the exception of the fifth postulate, which looks very complicated. That one is the parallel postulate, and that complicated appearance is what one gets when one tries to explain what "parallel" means.

Mathematicians tried for centuries to prove the fifth postulate from the others, without any success. Their proofs often involved sneaking in equivalent statements. But by the mid 19th cy., mathematicians Nikolai Lobachevsky and Janos Bolyai showed that rejecting it did not contradict the other axioms, thus creating non-Euclidean geometry.

 

[lpetrich]

Mathematicians over the last couple centuries have found many problems with Euclid's axioms, like incompleteness. Some of his proofs involved moving geometric figures, without explicitly stating that it is possible to do so.

A modern statement of axioms of equality is

1. Reflexivity: x = x
2. Symmetry: x = y implies y = x
3. Transitivity: x = y and y = z implies x = z

Here are Peano's axioms of the natural numbers. They involve numbers, zero, and a successor function S. They include the equality axioms with an additional one: closure. If x is a number and x = y, then y is also a number. Here goes:

1. 0 is a number
2. If x is a number, then S(x) is also a number.
3. If S(x) = S, then x = y, and in reverse.
4. There is no number x such that S(x) = 0
5. Mathematical induction: if there is a predicate function f such that f(0) is true, and for all numbers x, f(x) implies f(S(x)) is true, then f(x) is true for all x.

The mathematical-induction axiom can also be expressed with set membership. A predicate function returns a logical value: true or false.

One defines addition with

1. x + 0 = x
2. x + S = S(x + y)

multiplication with

1. x * 0 = 0
2. x * S = (x * y) + x

and ordering x <= y as true if there is some number z such that x + z = y.

Here is a modern statement of the axioms involved in Euclidean geometry. Without the fifth postulate, one has what is called absolute or neutral geometry. Its axioms, from Plane-Geometry Theorems

1. Existence: at least two points exist
2. Incidence: every line has some points. Two distinct points have a unique line between them.
3. Ruler: every pair of points on a line has a numerical distance value for them.
4. Plane Separation: every line L separates the plane into two convex regions, R1 and R2. If point P1 is in R1 and point P2 is in R2, then a line between P1 and P2 will intersect L.
5. Protractor: every angle has a numerical value.
6. Side-Angle-Side: for two triangles, ABC and DEF, if distance values (AB) = (DE) and (BC) = (EF), and if angle values (ABC) = (DEF), then the two triangles are congruent. All their corresponding distance and angle values are equal.
7. Area: two congruent triangles have the same area, and the area of two nonoverlapping regions is the sum of their areas.

For Euclidean geometry, one has in addition:

1. Parallel: the fifth one. For every line L and every point P not on it, there is exactly one line M that goes through P and nowhere intersects L.
2. Area: for a rectangle, the area is length * width.

 

[lpetrich]

The first form of non-Euclidean geometry recognized was hyperbolic geometry. Euclid's fifth is modified:

For every line L and point P not on it, there is more than one line on P that does not intersect L.

Then elliptic geometry. Euclid's fifth is modified:

For every line L and point P not on it, there is no line on P that does not intersect L.

Of the axioms of neutral geometry, the SAS postulate implies constant curvature. Elliptic geometry is thus the geometry of a sphere with each point identified with its antipode. Hyperbolic geometry is thus the geometry of one of the sheets of a two-sheet hyperboloid, though it is often projected into a disk.

-

Translated into arithmetic operations, straightedge-and-compass operations become basic arithmetic - addition, subtraction, multiplication, division - and taking square roots. This makes it possible to solve quadratic equations with real-number solutions using Euclid-style constructions.

There were three problems from antiquity that could not be solved with Euclid-style constructions: the duplication of the cube, the squaring of the circle, and the trisection of the angle.

Duplication of the cube: 2^(1/3)
like
Duplication of the square: 2^(1/2)

Squaring of the circle: sqrt(pi)

Trisection of the angle: x = cos(a/3), y = cos(a), 4*x^3 - 3*x = y
like
Bisection of the angle: x = cos(a/2), y = cos(a), 2*x^2 - 1 = y
Solution: x = sqrt(y + 1)

Duplication of the cube and trisection of the angle require solving cubic equations, unlike duplication of the square and bisection of the angle, which require solving quadratic equations, and are thus tractable with Euclid's methods.

But one duplicate the cube and trisect the angle with a neusis (marked straightedge) -- a neusis makes it possible to solve cubic and quartic equations with straightedge-and-compass constructions.

But the squaring of the circle is impossible with such constructions, even with a neusis. That is because pi is transcendental, solving no polynomial equation with integer coefficients.

 

[lpetrich]

An interesting problem is construction of a regular polygon. This is equivalent to finding trigonometric functions of angle (180d)/n for n sides. With trig identities, one gets integer-coefficient equations for those trig functions.

With Euclid-style methods, one can construct a regular polygon with n = 2^k * (product of distinct Fermat primes)

A Fermat prime is a Fermat number that is a prime: F(k) = 2^(2^k) + 1. Only five are known: F(0) = 3, F(1) = 5, F(2) = 17, F(3) = 257, F(4) = 65537. No others are known, up to F(32), and it is unknown whether or not there are any others.

With a neusis, one can solve cubic equations, and possible n = 2^k1 * 3^k2 * (product of distinct Pierpont primes)

Pierpoint primes have form 2^k1 * 3^k2 + 1

 

[lpetrich]

Now some videos.

The Three Square Geometry Problem - Numberphile - YouTube

Consider three squares in a row, from left to right. Consider square vertices on the left end: UL and DL. Consider the lower right vertices of each of the three squares: DR1, DR2, DR3.

Add the sizes of angles DL-DR1-UL, DL-DR2-UL, and DL-DR3-UL. Do they add up to 90d?

Zvezdelina Stankova does an ingeniously simple construction that gives the answer.

Squares & Triangles (EXTRA FOOTAGE #1) - YouTube

Mentions a triangles-in-a-square problem that goes back to Archimedes, the  Ostomachion

Squares & Triangles (EXTRA FOOTAGE #2) - YouTube

ZS mentioned at least 54 solutions for this problem. Mentions Charles Trigg, "A Three-Square Geometry Problem" - in the Journal of Recreational Mathematics

 

[Swammerdami]

With Euclid-style methods, one can construct a regular polygon with n = 2^k * (product of distinct Fermat primes)

The young Carl Gauss showed how to construct a 17-gon. (Did he ask that such a construction appear on his tombstone?) Wikipedia has a gif showing the construction — it isn't easy.

In the 2nd Proposition of Elements Euclid showed that any construction doable with straightedge and compass was still doable if the compass was collapsing (i.e. it could not be used directly to make a circle with center A and radius BC).

In the 18th century Mohr and Mascheroni showed that these constructions were doable even without a straightedge! (You wouldn't be able to show the lines, but could find all the relevant vertices and intersections.

Finally, in the 19th century, Steiner (building on work by Lambert and Poncelet) proved that any straightedge-and-compass construction could be done with straightedge alone(!) as long as the picture plane included any circle and its center.

 

[lpetrich]

A Miraculous Proof (Ptolemy's Theorem) - Numberphile - YouTube

Consider a quadrangle ABCD whose vertices lie on a circle. Ptolemy's theorem:
(AB) * (CD) + (AD) * (BC) = (AC) * (BD)

where (line segment) gives its length. It was stated by astronomer Claudius Ptolemy. Specializing to a rectangle gives Pythagoras's theorem.

Zvezdelina Stankova discusses a proof by circle inversion, and she adds a bit more in.
Inversion (extra) - Numberphile - YouTube

Pentagons and the Golden Ratio - Numberphile - YouTube

How one gets (sqrt(5) + 1)/2 for a regular pentagon's (diagonal length)/(side length) ratio. Ptolemy's theorem makes it easy.

 

[lpetrich]

Using the modern version of Euclidean-geometry axioms in Plane-Geometry Theorems I will try to derive them from analytic geometry, making notes about generalizations as I go.

-

First, the existence postulate.

Analytic geometry uses the concept of a vector space. Each point has an ordered tuple of D values, where D is the number of dimensions.

For Euclidean plane geometry, each point has two real numbers, and for Euclidean spatial geometry, three real numbers. One can extend this to arbitrarily many dimensions, making Euclidean geometry with arbitrary numbers of dimensions. One can generalize to limited regions, like 0 to 1 in each dimension, and one can use different kinds of values, like integers mod some number.

One can go even further, with an analogy between vector spaces and function spaces. A vector space is a function space where the functions' args range over the vector indices. But I'll leave that aside, but it's an indicator of how far mathematics has been developed over the last few centuries.

 

[lpetrich]

Now the incidence postulate.

In Euclidean geometry, a line between distinct points A and B has point x as a function of parameter t:

x = x(A) + (x(B) - x(A))*t

For t = 0, x = x(A) and for t = 1, x = x(B)

This expression is rather obviously unique.

Then the ruler postulate.

This involves a distance function. For points x and y,

dist(x,y) = sqrt( sum over i from 1 to D of (y(i) - x(i))^2 )

This is Pythagoras's theorem, and derivation of that theorem from Euclidean-geometry axioms shows that those axioms are consistent with this derivation from analytic geometry.

To get closer to the statement of the ruler postulate, we change the equation of the line:

x = x0 + n*t

where n.n = sum over i from 1 to D of n(i)^2 = 1

Finding t(A) from x(A) and t(B) for x(B) for points A and B on this line, we get

dist(x(A),x(B)) = |t(B) - t(A)|

as needed for this axiom.

I think that this discussion reveals a great weakness of Euclidean geometry. It seems to lack a concept of line direction.

By contrast, this formulation of lines has a built-in direction in each line, the direction of vector n.

If a line was defined with two points, we can nevertheless give that line a direction. Let (A) < (B) for definition points A and B, where the < is an ordering relation. For a point C distinct from A and B, then we have three possibilities:

dist(A,C) + dist(A,B) = dist(B,C) -- (C) < (A) < (B)
dist(A,C) + dist(B,C) = dist(A,B) -- (A) < (C) < (B)
dist(A,B) + dist(B,C) = dist(A,C) -- (A) < (B) < (C)

Thus placing every point on the line in that ordering, and thus giving the line a well-defined direction.

 

[lpetrich]

I'll now generalize Pythagoras's theorem. Use a D*D matrix g:

dist(x,y) = sqrt( sum over i,j from 1 to D of (y(i)-x(i)) * g(i,j) * (y(j)-x(j)) )

The matrix g can be made symmetric without loss of generality. One can rotate the coordinates so that they combine with g to make the result a diagonal matrix.

dist(x,y) = sqrt( sum over i from 1 to D of g(i) * (y(i)-x(i))^2 )

where the g(i) are the diagonal values. Staying in the real numbers, one can scale the dimensions so that the g(i)'s are +1, 0, or -1. For 0, the dimension drops out, and for all +1 and -1 we get the Euclidean distance measure again.

But for a mixture of +1's and -1's, we get something called Minkowskian geometry, after Hermann Minkowski, who discovered that special relativity implies a mixed-sign distance relation:

(distance)^2 = -c^2*(time)^2 = (x1)^2 + (x2)^2 + (x3)^2 - c^2*(t)^2

or more usually,

(distance)^2 = - (time)^2 = (x1)^2 + (x2)^2 + (x3)^2 - (t)^2

setting the speed of light in a vacuum c = 1, as is commonly done in theoretical work involving relativistic systems. The x's are space-dimension differences and the t a time-dimension difference. The speed of light is a speed that's built in to the geometry of space-time.

So time is a sort of imaginary distance, and distance a sort of imaginary time. There is a third kind of distance value, zero, and nonzero intervals can have it, thus being null intervals.

This also means that there are five kinds of space-time intervals:

• Forward timelike
• Forward null
• Spacelike
• Backward null
• Backward timelike

 

[lpetrich]

One can also make the matrix g, the "metric tensor", variable, but doing so means that one has to take the limit of bringing the two points together. One gets a differential form of the distance:

ds^2 = g_ij dxⁱ dx^j

using some common notational conventions in differential geometry. Repeated indices are summed over, raised indices are not powers but indicate that something varies like a coordinate, and lowered indices indicate that something varies like a gradient in the coordinates.

One has a concept of a straight line, something called a "geodesic". It's the line that makes the distance between two points the smallest or the largest. The latter can happen in Minkowskian geometries like what space-time has.

One can also find the curvature of the space, but it's a very complicated function of the metric tensor. The curvature itself is a multicomponent object, like the metric tensor, but more complicated. The "Riemann curvature tensor" is a four-index object, while the metric tensor is a two-index one, and coordinates are a one-index one.

 

[lpetrich]

Next is the plane separation axiom.

Consider a line x(t) = x0 + n*t, defined by x0 and n, with point parameter t

The region discriminator h = n.eps2.(x - x0)

where eps2 = {{0,1},{-1,0}}

The line thus divides the plane into two regions, one with h > 0 and one with h < 0

Consider some points x1 and x2, each one on each side of this line. For h1 = h(x1) and h2 = h(x2),
h1 < 0, h2 > 0 or h1 > 0, h2 < 0

Take a line between them and find the intersection with the region-dividing line. Solve

x1 + (x2 - x1)*u = x0 + n*t

Multiplying both sides by n.eps2 and rearranging, we get

h1 + (h2 - h1)*u = 0

giving u = h1/(h1 - h2)

Since h1 and h2 have opposite signs, u will be between 0 and 1, as one would expect.

 

[lpetrich]

One can generalize to more dimensions. Generalizing a line, we find a hyperplane:

x = x0 + n1*t1 + n2*t2 + ... + n(P)*t(P)

for a P-dimensional hyperplane. All the n's must be linearly independent. There must be no values of the t's that will make the n*t sum equal the zero vector.

0 dimensions: point
1 dimension: line
2 dimensions: plane
...

For a D-dimensional space, there is a version of the plane separation postulate. A (D-1)-dimensional hyperplane will separate the space into two regions, and a line from one region to the other will cross that hyperplane.

For this, we need to generalize the antisymmetric symbol eps2 to D dimensions: eps(D). It has D indices, and this value for them:
If all the index values are distinct, then:
- If they form an even permutation, then eps(indices) = 1
- If they form an odd permutation, then eps(indices) = -1
Otherwise, at least one index value is repeated, and eps(indices) = 0

A permutation is even or odd depending on whether it takes an even or an odd number of interchanges to make that permutation. Being even or odd is the permutation's "parity".

For one dimension, eps(1) = 1
For two dimensions, eps(1,1) = 0, eps(1,2) = 1, eps(2,1) = -1, eps(2,2) = 0
For three dimensions, eps(1,2,3) = eps(2,3,1) = eps(3,1,2) = 1, eps(1,3,2) = eps(2,1,3) = eps(3,2,1) = -1, all others = 0

The separation function is
h(x) = (n1, n2, ..., n(D-1)).eps(D).(x - x0)

and for line x1 + (x2-x1)*u with x1 and x2 on opposite sides of the hyperplane, the value of u is the same as before: h1/(h1 - h2), where h1 = h(x1) and h2 = h(x2)

 

[lpetrich]

I have been talking about Euclidean vector spaces, vector spaces that are R^D where R is the real numbers.

But one can depart with such things as finite domains, like R{0,1} - real numbers between 0 and 1 - and boundary conditions like periodic boundary conditions. Consider R{0,1}. Try an addition like 0.6 + 0.5 in it. Over plain R, one gets 1.1, but on R{0,1} it is undefined. But if we give periodic boundary conditions, then the value wraps around: 1.1 - 1 = 0.1, in R{0,1}.

Consider a rectangle with periodic boundary conditions in one dimension, let's say the horizontal one. Then a vertical line will seem to split this space into two regions, and I say "seem" because the boundary conditions mean that the region on one side will wrap around to the region in the other side. Thus, they are globally one region, even if they are locally two regions.


Turning to the protractor postulate, it indicates that angles have well-defined measures or sizes.

A line segment AB between points A and B is the part of the line that goes through those points that is between those points.

Take three points A, B, and C, with line segments AB and BC. At B is an angle between AB and BC. Its size is
arccos( (x(BA).x(BC)) / (|x(BA)|*|x(BC)|) )
where
x(BA) = x(A) - x(B), x(BC) = x(C) - x(B), |x| = sqrt(x.x)

That size has the appropriate addition properties, and it must be pointed out that angles have the same weakness in Euclidean geometry as distances on lines: lack of direction.

For distance metric g, these formulas become
arccos( (x(BA).g.x(BC)) / (|x(BA)|*|x(BC)|) )
where
|x| = sqrt(x.g.x)

For curves, angles are also well-defined, but for curves BA and BC, one must replace x(BA) and x(BC) with the tangent vectors of the curves at point B. That is also true of geodesics in curved spaces.

For a Minkowskian space, angle values with the above formula can become complex.

 

[lpetrich]

Now the side-angle-side postulate. It states for triangles ABC and DEF that if
length(AB) = length(DE)
length(BC) = length(EF)
angle-size(ABC) = angle-size(DEF)

then the triangles are congruent:
length(AC) = length(DF)
angle-size(BAC) = angle-size(EDF)
angle-size(BCA) = angle-size(EFD)

The SAS postulate follows from the translation (shift), rotation, and reflection symmetries of the space.

For a more general space, the SAS postulate implies constant curvature. This result is rather complicated to prove, requiring the full apparatus of differential geometry, and requiring using triangles relatively small compared to the curvature size scale, but I've been able to prove it.

For constant curvature K, the sum of the angles of a n-gon is
(n-2)*pi + K*(area)

For variable curvature, this result becomes
(n-2)*pi + (integral of K over area)

-

The final axiom of neutral or absolute geometry is the neutral area postulate.
- The areas of congruent triangles are equal
- The total area of two nonoverlapping polygonal regions is the sum of the areas of those regions.

For a Euclidean space, the area of a polygon is given in terms of its vertex coordinates by the shoelace formula:
(1/2)*sum over i of (x(i,1)*x(j,2) - x(i,2)*x(j,1)) or x(i).eps2.x(j)

For n vertices, j = i+1 for i < n, j = 1 for i = n.

This formula satisfies both parts of that postulate.

 

[lpetrich]

theorems-plane-geom.DVI - theorems-plane-geom.pdf
After listing several theorems that can be proved with these axioms, that document gets to the additional axioms for Euclidean geometry.

The parallel postulate, Euclid's fifth postulate: for a line L and point P not on it, there is exactly one line that goes through P and does not intersect L.

Proof: let us look for intersections of lines.
x1 + n1*t1 = x2 + n2*t2
Multiplying by n1.eps2 and n2.eps2 makes the calculation easy:

t1 = (n2.eps2.(x2 - x1)) / (n2.eps2.n1)
t2 = (n1.eps2.(x1 - x2)) / (n1.eps2.n2)

If the original line has direction n and the new one has direction n', the two lines not intersecting implies that n.eps2.n' = 0. This means that n' must be proportional to n. Since that proportionality can be absorbed into the position parameter, there is thus only one line that satisfies that condition.

So I have derived that troublesome axiom from analytic geometry and a manifestly flat space.


The remaining axiom is the Euclidean area postulate, that the area of a rectangle is the product of the lengths of opposite sides.

That's easy to obtain from the shoelace formula. For rectangle vertices (0,0), (a,0), (a,b), (0,b) I find
(1/2) * ( (0,0).eps2.(a,0) + (a,0).eps2.(a,b) + (a,b).eps2.(0,b) + (0,b).eps2.(0,0))
giving
(1/2) * (0 + a*b + a*b + 0) = a*b

 

[lpetrich]

An example of the incompleteness of Euclid's axioms is a theorem called the  Pons Asinorum, the Bridge of Donkeys. It likely got its name from being a difficult theorem for some learners.

Consider a triangle ABC. If sides AB and AC have the same lengths, then angles ABC and ACB have the same sizes. One needs the SAS postulate to prove it, and one uses an ingenious trick with it: flipping the triangle over. Map triangle ABC onto triangle ACB. Then B gets mapped onto C and C onto B. Angles BAC and CAB are the same angle, and thus have the same side. Sides AB and AC have the same length, and with SAS, angles ABC and ACB have equal size.


Turning to what polygons are constructible -  Constructible polygon - only a limited set can be constructed with a straightedge and compass. One can do bisection of a general angle, construction of certain angles that are (complete circle)/(prime), and combinations of these - and that's all. Neusis construction can do angle trisection (division by 3).  Tomahawk (geometry) is another geometry tool for making such constructions. It can supposedly be used in quinquisection / quintisection / pentasection (division by 5) of angles, but I have not been able to find any details of that.

That aside, I have calculated which polygons are constructible, working from some base set of primes.

Straightedge-and-compass (traditional Euclidean):

• Max number: 100
• Base primes: 2
• Fermat-like primes: 3, 5, 17
• Other primes: 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
• Constructible sizes: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96
• Other sizes: 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100

With a neusis or a tomahawk:

• Max number: 100
• Base primes: 2, 3
• Fermat-like primes: 3, 5, 7, 13, 17, 19, 37, 73, 97
• Other primes: 11, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89
• Constructible sizes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97
• Other sizes: 11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100

Possibly with a tomahawk:

• Max number: 100
• Base primes: 2, 3, 5
• Fermat-like primes: 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97
• Other primes: 23, 29, 43, 47, 53, 59, 67, 71, 79, 83, 89
• Constructible sizes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 88, 90, 91, 93, 95, 96, 97, 99, 100
• Other sizes: 23, 29, 43, 46, 47, 49, 53, 58, 59, 67, 69, 71, 79, 83, 86, 87, 89, 92, 94, 98

 

[lpetrich]

The use of lines and circles and similar tools in Euclidean constructions means that angles are not manipulated directly, but by way of their trigonometric functions. This may seem to make the problem of constructible angles very difficult to solve, since these functions are transcendental functions, but these functions are some identities that make the problem much simpler.

Let's consider multiple-angle formulas: cos(n*a) and sin(n*a) in terms of cos(a) and sin(a)

cos(n*a) = T(n,x) where x = cos(a)
sin(n*a) = U(n-1,x) * sin(a)

Let us use the trigonometric addition formulas to find the next members of the series:
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), sin(a+b) = sin(a)*cos(b) + sin(b)*cos(a)

This gives us
T(n+1,x) = x*T(n,x) - (1-x^2)*U(n-1,x)
U(n,x) = x*U(n-1,x) + T(n,x)

One can combine to make pure cosines or sines:
cos(b+a) + cos(b-a) = 2*cos(b)*cos(a)
sin(b+a) + sin(b-a) = 2*sin(b)*cos(a)

Thus,
T(n+1,x) - 2*x*T(n,x) + T(n-1,x) = 0
U(n+1,x) - 2*x*U(n,x) + U(n-1,x) = 0

The lowest members of these sequences are T(0,x) = 1, T(1,x) = x, U(0,x) = 1, U(1,x) = 2x, using sin(2a) = 2*sin(a)*cos(a)

This means that all the T(n,x)'s and U(n,x)'s are polynomials in x of degree n with integer coefficients -- the Chebyshev polynomials of the first and second kinds.

 

[lpetrich]

So the problem of the n-section of an angle reduces to solving a polynomial equation: T(n,cos(a/n)) = cos(a)

Bisection of an angle is easy: 2*(cos(a/2))^2 - 1 = cos(a)

This is a quadratic equation (degree 2), with solution
cos(a/2) = sqrt( (1 - cos(a)) / 2 )

Trisection requires solution of a cubic equation (degree 2), and in general, such equations are much more difficult to solve. In particular, their solution involves taking cube roots.

Quartic equations (degree 4) are even more difficult to solve, and on the way, they require solving cubic equations.

Quintic equations (degree 5) cannot, in general, be solved with nth roots.

BTW, degree 1 is linear.

Numbers that can be constructed with straightedge-and-compass constructions are called "constructible numbers". These are numbers that can be constructed from rational numbers by any finite combination of addition, subtraction, multiplication, division, and taking square roots.

With these techniques, one can solve linear and quadratic equations, but no higher-degree polynomial ones. With a neusis (marked straightedge), one can go up to cubic and quartic equations, and with a tomahawk geometric tool, one may be able to do quintic equations in addition to cubic and quartic ones.