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This pdf includes the following topics:-

Philosophy of Constructions

Compass vs. Dividers

Basic Constructions

Constructible Numbers

Constructions

Impossibility Proofs

Examples

Philosophy of Constructions

Compass vs. Dividers

Basic Constructions

Constructible Numbers

Constructions

Impossibility Proofs

Examples

1.
Geometric Constructions

2.
Philosophy of Constructions

Constructions using compass and straightedge have

a long history in Euclidean geometry. Their use

reflects the basic axioms of this system. However,

the stipulation that these be the only tools used in a

construction is artificial and only has meaning if

one views the process of construction as an

application of logic. In other words, this is not a

practical subject, if one is interested in constructing

a geometrical object there is no reason to limit

oneself as to which tools to use.

Constructions using compass and straightedge have

a long history in Euclidean geometry. Their use

reflects the basic axioms of this system. However,

the stipulation that these be the only tools used in a

construction is artificial and only has meaning if

one views the process of construction as an

application of logic. In other words, this is not a

practical subject, if one is interested in constructing

a geometrical object there is no reason to limit

oneself as to which tools to use.

3.
Philosophy of Constructions

The value of studying these constructions lies in

the rich supply of problems that can be posed in this

way. It is important that one be able to analyze a

construction to see why it works. It is not important

to gain the manual dexterity needed to carry out a

careful construction.

The value of studying these constructions lies in

the rich supply of problems that can be posed in this

way. It is important that one be able to analyze a

construction to see why it works. It is not important

to gain the manual dexterity needed to carry out a

careful construction.

4.
Compass vs. Dividers

The ancient Greek tool used to construct circles is

not the modern day compass. Rather, they used a

device known as a divider. Dividers consist of just

two arms with a central pivot. Should you pick up a

divider, the arms will collapse, so it is impossible to

use them to transfer lengths from one area to

another. Modern compasses remain open when

picked up, so such transfers are possible. Given the

difference in the two tools, it appears that the

modern compass is a more powerful instrument,

capable of doing more things.

The ancient Greek tool used to construct circles is

not the modern day compass. Rather, they used a

device known as a divider. Dividers consist of just

two arms with a central pivot. Should you pick up a

divider, the arms will collapse, so it is impossible to

use them to transfer lengths from one area to

another. Modern compasses remain open when

picked up, so such transfers are possible. Given the

difference in the two tools, it appears that the

modern compass is a more powerful instrument,

capable of doing more things.

5.
Compass vs. Dividers

However, this is not true.

The ancient dividers can do everything that modern

compasses can. Of course, this means that how

certain constructions were done by the ancient

Greeks are quite different from the way we would

do them today. This underscores the statement

above; technique is not as important as

understanding why it works.

However, this is not true.

The ancient dividers can do everything that modern

compasses can. Of course, this means that how

certain constructions were done by the ancient

Greeks are quite different from the way we would

do them today. This underscores the statement

above; technique is not as important as

understanding why it works.

6.
Basic Constructions

The basic constructions are:

1. Transfer a segment.

2. Bisect a line segment.

3. Construct a perpendicular to a line at a point on the line.

4. Construct a perpendicular to a line from a point not on

the line.

5. Construct an angle bisector.

6. Copy an angle.

7. Construct a parallel to a line through a given point.

8. Partition a segment into n congruent segments.

9. Divide a segment into a given ratio (internal and

The basic constructions are:

1. Transfer a segment.

2. Bisect a line segment.

3. Construct a perpendicular to a line at a point on the line.

4. Construct a perpendicular to a line from a point not on

the line.

5. Construct an angle bisector.

6. Copy an angle.

7. Construct a parallel to a line through a given point.

8. Partition a segment into n congruent segments.

9. Divide a segment into a given ratio (internal and

7.
Basic Construction 1

Transfer a line segment.

A'

A B

Transfer a line segment.

A'

A B

8.
Basic Construction 2

9.
Basic Construction 3

10.
Basic Construction 4

11.
Basic Construction 5

12.
Basic Construction 6

13.
Basic Construction 7

14.
Basic Construction 8

15.
Basic Construction 9

16.
Basic Construction 9

17.
Constructible Numbers

Given a segment which represents the number 1 (a unit segment),

the segments which can be constructed from this one by use of

compass and straightedge represent numbers called Constructible

Numbers. Note that the restrictions imply that the constructible

numbers are limited to lying in certain quadratic extensions of the

Given two constructible numbers one can with straightedge and

compass construct their:

Square Root

Given a segment which represents the number 1 (a unit segment),

the segments which can be constructed from this one by use of

compass and straightedge represent numbers called Constructible

Numbers. Note that the restrictions imply that the constructible

numbers are limited to lying in certain quadratic extensions of the

Given two constructible numbers one can with straightedge and

compass construct their:

Square Root

18.
Constructible Numbers

a

b

a + b

a

| |

b b–a

a

b

a + b

a

| |

b b–a

19.
Constructible Numbers

1

ab

a

b

b

1

a

1

a

a a/b

b 1 b

1

ab

a

b

b

1

a

1

a

a a/b

b 1 b

20.
Constructible Numbers

Square Root

1

a

a

a 1

Square Root

1

a

a

a 1

21.
Constructions

Example: Construct a triangle, given the length of one side of the

triangle, and the lengths of the altitude and median to that side.

Example: Construct a triangle, given the length of one side of the

triangle, and the lengths of the altitude and median to that side.

22.
Constructions

As the third vertex is determined by the intersection of one of two parallel

lines with a circle, there are three possibilities for the number of solutions. If b

is less than c, there will be no intersection, so no solutions. If b equals c, the

lines will be tangent to the circle and we would get two solutions. Finally, if b

is greater than c (the situation drawn above) then there will be four points of

As the third vertex is determined by the intersection of one of two parallel

lines with a circle, there are three possibilities for the number of solutions. If b

is less than c, there will be no intersection, so no solutions. If b equals c, the

lines will be tangent to the circle and we would get two solutions. Finally, if b

is greater than c (the situation drawn above) then there will be four points of

23.
Constructions

Example: Construct a triangle, given one angle, the length of the side opposite

this angle, and the length of the altitude to that side.

Example: Construct a triangle, given one angle, the length of the side opposite

this angle, and the length of the altitude to that side.

24.
Constructions

As the position of vertex A is determined by the intersection of a single line

with a circle, there are three possibilities for the number of solutions. If the

parallel does not intersect the circle, there is no solution. If the parallel is

tangent to the circle there is one solution, and finally, if the parallel intersects

the circle twice, there are two solutions (as indicated in the situation drawn

As the position of vertex A is determined by the intersection of a single line

with a circle, there are three possibilities for the number of solutions. If the

parallel does not intersect the circle, there is no solution. If the parallel is

tangent to the circle there is one solution, and finally, if the parallel intersects

the circle twice, there are two solutions (as indicated in the situation drawn

25.
Constructions

Example: Construct a triangle, given the circumcenter O, the center of the

nine-point circle N, and the midpoint of one side A'.

Example: Construct a triangle, given the circumcenter O, the center of the

nine-point circle N, and the midpoint of one side A'.

26.
Constructions

This construction always gives a unique triangle provided one exists. If N = A'

there will be no nine-point circle, but N could equal O, or A' could equal O

and the construction will still work. The points could also be collinear.

This construction always gives a unique triangle provided one exists. If N = A'

there will be no nine-point circle, but N could equal O, or A' could equal O

and the construction will still work. The points could also be collinear.

27.
Impossibility Proofs

An algebraic analysis of the fields of constructible

numbers shows the following:

Theorem: If a constructible number is a root of a cubic

equation with rational coefficients, then the equation must

have at least one rational root.

While we will not prove this result, we shall use it to

investigate some old geometric problems that dealt with

An algebraic analysis of the fields of constructible

numbers shows the following:

Theorem: If a constructible number is a root of a cubic

equation with rational coefficients, then the equation must

have at least one rational root.

While we will not prove this result, we shall use it to

investigate some old geometric problems that dealt with

28.
Impossibility Proofs

The three famous problems of antiquity are:

The Delian problem - duplicating the cube. The problem

is to construct a cube that has twice the volume of a given

cube. A particular instance of this problem would be to

construct a cube whose volume is twice that of the unit

cube. This entails constructing a side of the larger cube,

and in this case that means constructing a length equal to

the cube root of 2. This length is a root of the equation

3

x - 2 = 0, but this cubic equation with rational coefficients

has no rational root.

The three famous problems of antiquity are:

The Delian problem - duplicating the cube. The problem

is to construct a cube that has twice the volume of a given

cube. A particular instance of this problem would be to

construct a cube whose volume is twice that of the unit

cube. This entails constructing a side of the larger cube,

and in this case that means constructing a length equal to

the cube root of 2. This length is a root of the equation

3

x - 2 = 0, but this cubic equation with rational coefficients

has no rational root.

29.
Impossibility Proofs

Trisection of an Angle - The problem is to find the angle

trisectors for an arbitrary angle. The general problem can

not be done because it can't be done for some specific

angles, for instance an angle of 60º. (Construction of a 20

3

degree angle leads to the cubic equation 8x -6x - 1 = 0,

and this does not have roots of the required type).

Trisection of an Angle - The problem is to find the angle

trisectors for an arbitrary angle. The general problem can

not be done because it can't be done for some specific

angles, for instance an angle of 60º. (Construction of a 20

3

degree angle leads to the cubic equation 8x -6x - 1 = 0,

and this does not have roots of the required type).

30.
Impossibility Proofs

Squaring the Circle - The problem is to construct a square

that has the same area as the unit circle, i.e. π. If this can

be done, then the square root of π would be constructible.

And if that is true, then π would also be constructible. But

π is a transcendental number (Lindemann, 1882), and

such numbers are not constructible.

Squaring the Circle - The problem is to construct a square

that has the same area as the unit circle, i.e. π. If this can

be done, then the square root of π would be constructible.

And if that is true, then π would also be constructible. But

π is a transcendental number (Lindemann, 1882), and

such numbers are not constructible.

31.
Angle Trisection

Angle Trisection can be done in many ways, some of

which were known to the ancient Greeks. A simple

method which uses a marked straightedge is due to

Archimedes (287-212 B.C.) and another uses the

Conchoid of Nichomedes (240 B.C.).

Angle Trisection can be done in many ways, some of

which were known to the ancient Greeks. A simple

method which uses a marked straightedge is due to

Archimedes (287-212 B.C.) and another uses the

Conchoid of Nichomedes (240 B.C.).

32.
Archimedes' Angle Trisection

33.
Archimedes' Angle Trisection

B T

S

O A

Let ∠AOT = x. ∠AOT ≅ ∠OTB (alternate interior angles of || lines.)

∠OTB ≅ ∠TBS since ∆SBT is isosceles. ∠BSO = 2x since it is an

exterior angle which is equal to the sum of the two opposite interior

angles. ∠BOS ≅ ∠BSO since ∆BSO is isosceles. Therefore, ∠AOT is

1/3 of ∠AOB.

B T

S

O A

Let ∠AOT = x. ∠AOT ≅ ∠OTB (alternate interior angles of || lines.)

∠OTB ≅ ∠TBS since ∆SBT is isosceles. ∠BSO = 2x since it is an

exterior angle which is equal to the sum of the two opposite interior

angles. ∠BOS ≅ ∠BSO since ∆BSO is isosceles. Therefore, ∠AOT is

1/3 of ∠AOB.

34.
Conchoid of Nicomedes

Given a point O, a line l not through O and a length k we

form the conchoid by adding the length k to all line

segments drawn from O to l.

l

k

O

Given a point O, a line l not through O and a length k we

form the conchoid by adding the length k to all line

segments drawn from O to l.

l

k

O

35.
Conchoid of Nicomedes

B

C

k/2

A k/2

B

C

k/2

A k/2

36.
Circle Squarers

“We have not placed in the above chronology of π any items from the

vast literature supplied by sufferers of morbus cyclometricus, the circle-

squaring disease. These contributions, often amusing and at times almost

unbelievable, would require a publication all to themselves.”

- Howard Eves, An Introduction to the History of Mathematics

Circle squarers, angle trisectors, and cube duplicators are members

of a curious social phenomenon that has plagued mathematicians

since the earliest days of the science. They are generally older

gentlemen who are mathematical amateurs (although some have had

mathematical training) that upon hearing that something is

impossible are driven by some inner compulsion to prove the

authorities wrong.

“We have not placed in the above chronology of π any items from the

vast literature supplied by sufferers of morbus cyclometricus, the circle-

squaring disease. These contributions, often amusing and at times almost

unbelievable, would require a publication all to themselves.”

- Howard Eves, An Introduction to the History of Mathematics

Circle squarers, angle trisectors, and cube duplicators are members

of a curious social phenomenon that has plagued mathematicians

since the earliest days of the science. They are generally older

gentlemen who are mathematical amateurs (although some have had

mathematical training) that upon hearing that something is

impossible are driven by some inner compulsion to prove the

authorities wrong.

37.
Circle Squarers

In 1872, Augustus De Morgan's (1806-1871) widow edited and had

published some notes that De Morgan had been preparing for a

book, called A Budget of Paradoxes. A logician and teacher, De

Morgan had been the first chair in mathematics of London

University (from 1828). Besides his mathematical work, he wrote

many reviews and expository articles and much on teaching

mathematics. In the Budget, he examines his personal library and

satirically barbs all the examples of weird and crackpot theories that

he finds there. As he points out, these are just books that randomly

came into his possession – he did not seek out any of this type of

material. In the approximately 150 works he examined, there can be

found 24 circle squarers and an additional 19 bogus values of π.

In 1872, Augustus De Morgan's (1806-1871) widow edited and had

published some notes that De Morgan had been preparing for a

book, called A Budget of Paradoxes. A logician and teacher, De

Morgan had been the first chair in mathematics of London

University (from 1828). Besides his mathematical work, he wrote

many reviews and expository articles and much on teaching

mathematics. In the Budget, he examines his personal library and

satirically barbs all the examples of weird and crackpot theories that

he finds there. As he points out, these are just books that randomly

came into his possession – he did not seek out any of this type of

material. In the approximately 150 works he examined, there can be

found 24 circle squarers and an additional 19 bogus values of π.

38.
Angle Trisectors

DeMorgan's book was very successful. Today, with a couple of

notable exceptions, there are hardly any circle squarers left.

However, their cousins, the Angle Trisectors are still with us.

Underwood Dudley, in 1987, wrote A Budget of Trisections in an

attempt to do for Angle Trisectors what DeMorgan had done for

Circle Squarers.

The comments and quotes that follow are all from Dudley's book.

(The 2nd edition came out in 1996 and was renamed The

DeMorgan's book was very successful. Today, with a couple of

notable exceptions, there are hardly any circle squarers left.

However, their cousins, the Angle Trisectors are still with us.

Underwood Dudley, in 1987, wrote A Budget of Trisections in an

attempt to do for Angle Trisectors what DeMorgan had done for

Circle Squarers.

The comments and quotes that follow are all from Dudley's book.

(The 2nd edition came out in 1996 and was renamed The

39.
Angle Trisectors

There are several characteristics of angle trisectors (shared by others

of their ilk) that may help you identify them.

1. They are men. Almost universally. Women seem to have more

sense.

2. They are old. Often retired, having led a successful life in their

chosen endeavors. Too much free time.

3. They fail to understand what “impossible” means in

mathematics. The meaning is unfortunately not the same as the

meaning in English. It is one of the great failures of mathematics

education that this essential difference is not made plain to

students.

There are several characteristics of angle trisectors (shared by others

of their ilk) that may help you identify them.

1. They are men. Almost universally. Women seem to have more

sense.

2. They are old. Often retired, having led a successful life in their

chosen endeavors. Too much free time.

3. They fail to understand what “impossible” means in

mathematics. The meaning is unfortunately not the same as the

meaning in English. It is one of the great failures of mathematics

education that this essential difference is not made plain to

students.

40.
Angle Trisectors

Typical is the trisector who wrote

“I received through the mail an advertising brochure, from a

science magazine, that had in it a simple statement – and it went

something like this – the FORMULA for TRISECTING AN

ANGLE had never been worked out. This really intrigued me. I

couldn't believe, after hundreds of years of math, that this could

be true.”

So he went to the library and found that all the books agreed that

it was impossible.

“How could men of science be so stupid? Any scientist or

mathematician who declares that a thing is impossible is showing

his limitations before he even starts on the problem at hand.”

Typical is the trisector who wrote

“I received through the mail an advertising brochure, from a

science magazine, that had in it a simple statement – and it went

something like this – the FORMULA for TRISECTING AN

ANGLE had never been worked out. This really intrigued me. I

couldn't believe, after hundreds of years of math, that this could

be true.”

So he went to the library and found that all the books agreed that

it was impossible.

“How could men of science be so stupid? Any scientist or

mathematician who declares that a thing is impossible is showing

his limitations before he even starts on the problem at hand.”

41.
Angle Trisectors

Another trisector wrote in 1933:

“Moreover, we find our modern authorities of mathematics not

attempting to solve these unsolved problems, but writing

treatises showing the impossibility of proving them. Instead of

offering inducements to the solution of these problems, they

discourage others and dub them as 'cranks'. “

4. They do not know much mathematics. Often, high school is

the last place they have seen any formal mathematics.

“It was necessary to get outside of the problem to solve it, and

it was not solved by a study of geometry and trigonometry, as

the author has never made a study of these branches of learning.”

Another trisector wrote in 1933:

“Moreover, we find our modern authorities of mathematics not

attempting to solve these unsolved problems, but writing

treatises showing the impossibility of proving them. Instead of

offering inducements to the solution of these problems, they

discourage others and dub them as 'cranks'. “

4. They do not know much mathematics. Often, high school is

the last place they have seen any formal mathematics.

“It was necessary to get outside of the problem to solve it, and

it was not solved by a study of geometry and trigonometry, as

the author has never made a study of these branches of learning.”

42.
Angle Trisectors

5. They think the problem is important. Since Archimedes work,

there has not been any need for such a construction, yet they

persist in thinking that mathematics has been stymied by this

lack.

“It having been hitherto deemed impossible to geometrically

trisect or divide any angle into any number of equal parts, or

fractions of parts, the author of the present work has devoted

careful study to the solving of the problem so useful and necessary

to every branch of science and art, that requires the use of

“The study of technical magazines and data shows that a

solution is being sought whereby a standard construction permits

the thrice division of any given angle ...”

5. They think the problem is important. Since Archimedes work,

there has not been any need for such a construction, yet they

persist in thinking that mathematics has been stymied by this

lack.

“It having been hitherto deemed impossible to geometrically

trisect or divide any angle into any number of equal parts, or

fractions of parts, the author of the present work has devoted

careful study to the solving of the problem so useful and necessary

to every branch of science and art, that requires the use of

“The study of technical magazines and data shows that a

solution is being sought whereby a standard construction permits

the thrice division of any given angle ...”

43.
Angle Trisectors

6. They believe that they will be richly rewarded for their work. No

one has ever put up a prize for a solution.

“When the time came for me to submit this project to a

publisher, I was very much concerned about the copyright. I was

fearful that if I submitted to a publisher, they might steal the entire

trisection and I would have to go to court and try to establish my

right to the trisection.”

6. They believe that they will be richly rewarded for their work. No

one has ever put up a prize for a solution.

“When the time came for me to submit this project to a

publisher, I was very much concerned about the copyright. I was

fearful that if I submitted to a publisher, they might steal the entire

trisection and I would have to go to court and try to establish my

right to the trisection.”

44.
Angle Trisectors

7. They are not logical. For instance,

“Those who are skeptical should offer something more than

rhetoric or argument in order to disprove geometrical facts.

Assuming the angle and its trisectors given, the enveloping

quadrantal arc constructed, and its points of trisection found, if it be

denied that the trisectors pass through these points of equal division

on the quadrantal arc, let them show by the ruler and compasses

where these lines and points are with respect to each other on the

quadrant. If the lines constituting the respective pairs of trisectors

of both sectors do not intersect on the quadrantal arc they should

show by the ruler and compasses where they do intersect. ”

To prove him wrong you have to trisect an angle with ruler and

compass. !!!!

7. They are not logical. For instance,

“Those who are skeptical should offer something more than

rhetoric or argument in order to disprove geometrical facts.

Assuming the angle and its trisectors given, the enveloping

quadrantal arc constructed, and its points of trisection found, if it be

denied that the trisectors pass through these points of equal division

on the quadrantal arc, let them show by the ruler and compasses

where these lines and points are with respect to each other on the

quadrant. If the lines constituting the respective pairs of trisectors

of both sectors do not intersect on the quadrantal arc they should

show by the ruler and compasses where they do intersect. ”

To prove him wrong you have to trisect an angle with ruler and

compass. !!!!

45.
Angle Trisectors

8. They are loners. They work by themselves, sometimes using

books, but never discuss their work until it is completed. Even

though they do not communicate with each other, they do tend to

swarm.

For instance, in 1754, Jean Étienne Montucla, an early French

historian of mathematics, wrote a legitimate history of the

quadrature problem. A year later, the French Academy of

Sciences was forced to publicly announce that it would no longer

examine any solutions of the quadrature problem.

8. They are loners. They work by themselves, sometimes using

books, but never discuss their work until it is completed. Even

though they do not communicate with each other, they do tend to

swarm.

For instance, in 1754, Jean Étienne Montucla, an early French

historian of mathematics, wrote a legitimate history of the

quadrature problem. A year later, the French Academy of

Sciences was forced to publicly announce that it would no longer

examine any solutions of the quadrature problem.

46.
Angle Trisectors

9. They are prolific writers. Here is what De Morgan says about

Milan whose method gave π = 3.2 in 1855:

[The circle-squarer] is active and able, with nothing wrong with him

except his paradoxes. In the second tract named he has given the

testimonials of crowned heads and ministers, etc. as follows. Louis

Napoleon gives thanks. The minister at Turin refers it to the Academy of

Sciences and hopes so much labor will be judged worthy of esteem. The

Vice-Chancellor of Oxford – a blunt Englishman – begs to say that the

University has never proposed the problem, as some affirm. The Prince

Regent of Baden has received the work with lively interest. The

Academy of Vienna is not in a position to enter into the question. The

Academy of Turin offers the most distinct thanks. The Academy della

Crusca attends only to literature, but gives thanks. The Queen of Spain

has received the work with the highest appreciation. The University of

Salamanca gives infinite thanks, and feels true satisfaction in having the

9. They are prolific writers. Here is what De Morgan says about

Milan whose method gave π = 3.2 in 1855:

[The circle-squarer] is active and able, with nothing wrong with him

except his paradoxes. In the second tract named he has given the

testimonials of crowned heads and ministers, etc. as follows. Louis

Napoleon gives thanks. The minister at Turin refers it to the Academy of

Sciences and hopes so much labor will be judged worthy of esteem. The

Vice-Chancellor of Oxford – a blunt Englishman – begs to say that the

University has never proposed the problem, as some affirm. The Prince

Regent of Baden has received the work with lively interest. The

Academy of Vienna is not in a position to enter into the question. The

Academy of Turin offers the most distinct thanks. The Academy della

Crusca attends only to literature, but gives thanks. The Queen of Spain

has received the work with the highest appreciation. The University of

Salamanca gives infinite thanks, and feels true satisfaction in having the

47.
Angle Trisectors

Lord Palmerston gives thanks. The Viceroy of Egypt, not yet being

up in Italian, will spend his first moments of leisure in studying the

book, when it shall have been translated into French: in the mean

time he congratulates the author upon his victory over a problem so

long held insoluble. All this is seriously published as a rate in aid of

demonstration. If those royal compliments cannot make the

circumference about 2 per cent larger than geometry will have it –

which is all that is wanted – no wonder that thrones are shaky.

- Budget, Vol. 2, pp. 61-2.

Lord Palmerston gives thanks. The Viceroy of Egypt, not yet being

up in Italian, will spend his first moments of leisure in studying the

book, when it shall have been translated into French: in the mean

time he congratulates the author upon his victory over a problem so

long held insoluble. All this is seriously published as a rate in aid of

demonstration. If those royal compliments cannot make the

circumference about 2 per cent larger than geometry will have it –

which is all that is wanted – no wonder that thrones are shaky.

- Budget, Vol. 2, pp. 61-2.

48.
Angle Trisectors

Now, will you know a Angle Trisector when you see

one coming? And will you know what to do?

Hint: What you do involves your legs.

No, you do not kick him!

Now, will you know a Angle Trisector when you see

one coming? And will you know what to do?

Hint: What you do involves your legs.

No, you do not kick him!

49.
Regular Polygons (Gauss)

These are only possible when the number of sides, n, is of

the form

a

n = 2 p1p2...pk

where the pi are distinct Fermat primes, i.e. prime numbers

of the form 2

i

p i1 := 2 1

The first few Fermat primes are: p 1= 3, p2= 5, p3 = 17.

Thus, it is possible to construct regular polygons of n sides

2 3

when n is: 3, 4 = 2 , 5, 6 = 2(3), 8 = 2 , 10 = 2(5), 12 =

2 4

2 (3), 15 = 3(5), 16 = 2 and 17.

These are only possible when the number of sides, n, is of

the form

a

n = 2 p1p2...pk

where the pi are distinct Fermat primes, i.e. prime numbers

of the form 2

i

p i1 := 2 1

The first few Fermat primes are: p 1= 3, p2= 5, p3 = 17.

Thus, it is possible to construct regular polygons of n sides

2 3

when n is: 3, 4 = 2 , 5, 6 = 2(3), 8 = 2 , 10 = 2(5), 12 =

2 4

2 (3), 15 = 3(5), 16 = 2 and 17.

50.
Regular Polygons (Gauss)

The factor of a power of 2 comes from the fact that given

any regular n-gon, you can always construct a regular 2n-

gon. This is done by inscribing the n-gon in a circle and

then constructing the perpendicular bisectors of each of the

sides. Extend these to the circle and these points together

with the original vertices of the n-gon, form the vertices of

a regular 2n-gon. Repeating this will give the higher

powers of 2.

It is not possible to construct, with straightedge and

compass alone, regular polygons of sides n = 7, 9, 11, 13,

14, 18, 19, ....

The factor of a power of 2 comes from the fact that given

any regular n-gon, you can always construct a regular 2n-

gon. This is done by inscribing the n-gon in a circle and

then constructing the perpendicular bisectors of each of the

sides. Extend these to the circle and these points together

with the original vertices of the n-gon, form the vertices of

a regular 2n-gon. Repeating this will give the higher

powers of 2.

It is not possible to construct, with straightedge and

compass alone, regular polygons of sides n = 7, 9, 11, 13,

14, 18, 19, ....

51.
Other types of Constructions

It can be shown that any construction that can be made

with straightedge and compass can be made with compass

alone (Mascheroni, 1797 [Mohr, 1672]). Of course one

must understand that a straight line is given as soon as

two points on it are determined, since one can't actually

draw a straight line with only a compass.

It can also be shown that any construction that can be

made with straightedge and compass can be made with

straightedge alone, as long as there is a single circle with

its center given (Steiner, 1833 [Poncelet, 1822]).

It can be shown that any construction that can be made

with straightedge and compass can be made with compass

alone (Mascheroni, 1797 [Mohr, 1672]). Of course one

must understand that a straight line is given as soon as

two points on it are determined, since one can't actually

draw a straight line with only a compass.

It can also be shown that any construction that can be

made with straightedge and compass can be made with

straightedge alone, as long as there is a single circle with

its center given (Steiner, 1833 [Poncelet, 1822]).

52.
Paper Folding

53.
Paper Folding

54.
Paper Folding

Other constructions are possible, including trisecting angles.

Here is a folding of the regular nonagon (9 sided regular polygon) which is

impossible to do with straightedge and compass (from T.S. Row, Geometric

Exercises in Paper Folding).

Unfortunately, this is

only approximate, this

construction can not be

done exactly with paper

Other constructions are possible, including trisecting angles.

Here is a folding of the regular nonagon (9 sided regular polygon) which is

impossible to do with straightedge and compass (from T.S. Row, Geometric

Exercises in Paper Folding).

Unfortunately, this is

only approximate, this

construction can not be

done exactly with paper