Background The Construction.
Straight-Edge and Compass: Constructing the Heptadecagon
Daniel Baron
Western Washington University
May 15, 2013
Daniel Baron Construction of the Heptadecagon Background The Construction. Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon For example, here is a construction of an equilateral triangle:
Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius.
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History What Is Straightedge-And-Compass Construction?
It is a method for drawing “perfect” lengths and shapes with simple tools: An infinite, unruled straightedge. A compass which can draw a perfect circle of any radius. For example, here is a construction of an equilateral triangle:
Daniel Baron Construction of the Heptadecagon And most importantly, it’s fun to do!
Background Straightedge-And-Compass Construction The Construction. History Applications. Why bother?
We can use geometry to prove that a construction is exactly correct. This is significant in many fields. In particular, consider. . . Drafting, both by hand and with CAD software. The science of weights and measures.
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History Applications. Why bother?
We can use geometry to prove that a construction is exactly correct. This is significant in many fields. In particular, consider. . . Drafting, both by hand and with CAD software. The science of weights and measures. And most importantly, it’s fun to do!
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon Division of line segments into equal parts. Bisection of angles. Construction of parallel and perpendicular lines. Construction of regular 3-, 4-, 5-, and 15-gons. But very little progress was made after Euclid. In particular, no new polygon constructions were found until . . .
Background Straightedge-And-Compass Construction The Construction. History The Ancients.
The ancient Greeks were very skilled with compass constructions. Euclid detailed a great many, including . . .
Daniel Baron Construction of the Heptadecagon But very little progress was made after Euclid. In particular, no new polygon constructions were found until . . .
Background Straightedge-And-Compass Construction The Construction. History The Ancients.
The ancient Greeks were very skilled with compass constructions. Euclid detailed a great many, including . . . Division of line segments into equal parts. Bisection of angles. Construction of parallel and perpendicular lines. Construction of regular 3-, 4-, 5-, and 15-gons.
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History The Ancients.
The ancient Greeks were very skilled with compass constructions. Euclid detailed a great many, including . . . Division of line segments into equal parts. Bisection of angles. Construction of parallel and perpendicular lines. Construction of regular 3-, 4-, 5-, and 15-gons. But very little progress was made after Euclid. In particular, no new polygon constructions were found until . . .
Daniel Baron Construction of the Heptadecagon Gauss further proved that the regular n-gon is constructible if and only if n is of the form
r n = 2 p1p2 ... ps,
where r ≥ 0 and each pi is a distinct Fermat prime; 2k i.e., a prime of the form pi = 2 + 1. The known Fermat primes are 3, 5, 17, 257, and 65537.
Background Straightedge-And-Compass Construction The Construction. History The 18th Century and Beyond.
In 1796, Gauss discovered that the regular heptadecagon is constructible by ruler and compass.
Daniel Baron Construction of the Heptadecagon The known Fermat primes are 3, 5, 17, 257, and 65537.
Background Straightedge-And-Compass Construction The Construction. History The 18th Century and Beyond.
In 1796, Gauss discovered that the regular heptadecagon is constructible by ruler and compass. Gauss further proved that the regular n-gon is constructible if and only if n is of the form
r n = 2 p1p2 ... ps,
where r ≥ 0 and each pi is a distinct Fermat prime; 2k i.e., a prime of the form pi = 2 + 1.
Daniel Baron Construction of the Heptadecagon Background Straightedge-And-Compass Construction The Construction. History The 18th Century and Beyond.
In 1796, Gauss discovered that the regular heptadecagon is constructible by ruler and compass. Gauss further proved that the regular n-gon is constructible if and only if n is of the form
r n = 2 p1p2 ... ps,
where r ≥ 0 and each pi is a distinct Fermat prime; 2k i.e., a prime of the form pi = 2 + 1. The known Fermat primes are 3, 5, 17, 257, and 65537.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Or an angle:
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Bisection.
We can use compasses to bisect a given line segment:
Or an angle:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon A Time-Saving Tip.
It is usually only necessary to draw short arcs of your circles:
But perhaps it is not as visually pleasing.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Perpendicular lines.
Given a line and a point, we can construct a perpendicular line:
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
2π 5
Daniel Baron Construction of the Heptadecagon 2π 5
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
Daniel Baron Construction of the Heptadecagon 2π 5
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Polygons and Angles.
Suppose we wish to construct the regular pentagon. It turns out that this is equivalent to constructing the angle 2π/5.
Daniel Baron Construction of the Heptadecagon We can understand this in group theory terms, noting that any nonzero element k of the group Zn of integers mod n generates the whole group, so the sets
{0, k, 2k,..., (n − 1)k} = {0, 1, 2,..., n − 1}
are equal under mod n arithmetic.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Primitive Roots of Primes.
In fact, if n is prime, we can draw the n-gon given any angle 2kπ n , with k an integer and 0 < k < n. We can find each of the n 2kπ vertices of the polygon by copying n around a circle as we did 2π on the previous slide with 5 .
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Primitive Roots of Primes.
In fact, if n is prime, we can draw the n-gon given any angle 2kπ n , with k an integer and 0 < k < n. We can find each of the n 2kπ vertices of the polygon by copying n around a circle as we did 2π on the previous slide with 5 . We can understand this in group theory terms, noting that any nonzero element k of the group Zn of integers mod n generates the whole group, so the sets
{0, k, 2k,..., (n − 1)k} = {0, 1, 2,..., n − 1}
are equal under mod n arithmetic.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Primitive Roots of Primes.
Careful! This is only guaranteed if n is prime. In general, it works if gcd(k, n) = 1.
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π cos 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π2π coscos5 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
Daniel Baron Construction of the Heptadecagon 2π cos 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Daniel Baron Construction of the Heptadecagon 2π cos 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Daniel Baron Construction of the Heptadecagon 2π cos 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Daniel Baron Construction of the Heptadecagon 2π cos 5
Of course, the sine is also sufficient.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Daniel Baron Construction of the Heptadecagon 2π cos 5
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Sines and Cosines.
Constructing the cosine of an angle (i.e., constructing a line segment of that length) is just as good as constructing the angle itself:
2π cos 5
Of course, the sine is also sufficient.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Outline
1 Background Straightedge-And-Compass Construction History
2 The Construction. Basic Tools and Techniques. Constructing Polygons. How To Draw a Heptadecagon
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon A Brief Acknowledgment.
The construction method I will demonstrate today is due to Herbert William Richmond, who published it in 1893. I have only seen a few techniques, but I find his to be very elegant and I am proud to share it with you today.
Daniel Baron Construction of the Heptadecagon For convenience, denote α = 2π/17, so that we want cos 3α and cos 5α. Let’s see what they look like . . .
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Getting Started.
As we have seen, constructing the 17-gon is equivalent to 2kπ constructing the length cos 17 for any 0 < k < 17. In particular, today we will directly construct 6π cos 17 10π and cos 17 .
Daniel Baron Construction of the Heptadecagon Let’s see what they look like . . .
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Getting Started.
As we have seen, constructing the 17-gon is equivalent to 2kπ constructing the length cos 17 for any 0 < k < 17. In particular, today we will directly construct 6π cos 17 10π and cos 17 . For convenience, denote α = 2π/17, so that we want cos 3α and cos 5α.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Getting Started.
As we have seen, constructing the 17-gon is equivalent to 2kπ constructing the length cos 17 for any 0 < k < 17. In particular, today we will directly construct 6π cos 17 10π and cos 17 . For convenience, denote α = 2π/17, so that we want cos 3α and cos 5α. Let’s see what they look like . . .
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big, Complicated Numbers.
√ 1 p We have cos 3α = 4096 ( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ √ 3 3 p + 17 − 1) + 4096 (( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ p √ + 17 − 1)2 − 256)( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ + 17 − 1).
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big, Complicated Numbers. √ 1 p And cos 5α = 1048576 ( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ √ 5 5 p + 17 − 1) + 524288 (( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ p √ + 17 − 1)2 − 256)( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ √ 3 5 p + 17 − 1) + 1048576 (( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ p √ + 17 − 1)2 − 256)2( −2 17 + 34 q p √ √ p √ √ + 2 −2 17 + 34( 17 − 1) − 16 2 17 + 34 + 12 17 + 68 √ + 17 − 1).
Daniel Baron Construction of the Heptadecagon It can be shown that
tan φ = 2(cos 3α + cos 5α)
and π tan(φ − ) = 4 cos 3α cos 5α. 4 We will now construct these two lengths.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Easy Way
Consider the acute angle φ such that tan 4φ = 4 (so φ is about 18◦).
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Easy Way
Consider the acute angle φ such that tan 4φ = 4 (so φ is about 18◦). It can be shown that
tan φ = 2(cos 3α + cos 5α)
and π tan(φ − ) = 4 cos 3α cos 5α. 4 We will now construct these two lengths.
Daniel Baron Construction of the Heptadecagon B
Bisect OB twice to find I at I 1 (0, 4 ). Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. Next, find a perpendicular radius.
A O
Daniel Baron Construction of the Heptadecagon B
Bisect OB twice to find I at I 1 (0, 4 ). Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. Next, find a perpendicular radius.
A O
Daniel Baron Construction of the Heptadecagon B
Bisect OB twice to find I at I 1 (0, 4 ). Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. Next, find a perpendicular radius.
A O
Daniel Baron Construction of the Heptadecagon Bisect OB twice to find I at I 1 (0, 4 ). Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius.
A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon I Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon Using your straightedge, draw the line segment AI.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at I 1 (0, 4 ). A O
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at I 1 (0, 4 ). A Using your straightedge, draw O the line segment AI.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon First Steps.
Start with a unit circle and arbi- trary diameter. The point A is your 0th vertex. B Next, find a perpendicular radius. Bisect OB twice to find I at I 1 (0, 4 ). A Using your straightedge, draw O the line segment AI.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle 4φ.
B Recall that φ is acute and that tan 4φ = 4. Since I 1 tan ∠OIA = OA/OI 4 A O 1 = 4,
we have already constructed the angle 4φ.
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact, E tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A O
Daniel Baron Construction of the Heptadecagon There it is, φ = ∠OIE. 1 4 And in fact,
tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I
A OE
Daniel Baron Construction of the Heptadecagon 1 4 And in fact,
tan φ = tan ∠OIE = OE/OI = 4 OE.
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I There it is, φ = ∠OIE. A OE
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Angle φ and Its Tangent.
B Next, bisect ∠OIA twice to con- struct φ. I There it is, φ = ∠OIE. 1 And in fact, 4 A OE tan φ = tan ∠OIE = OE/OI = 4 OE.
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I.
A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . F And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A OE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . . And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A FOE
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . .
Further, notice that π tan(φ − ) = 4OF. 4
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A FOE And you have it.
Daniel Baron Construction of the Heptadecagon Next, bisect that right angle . . .
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon π The Angle φ − 4 .
B π To construct φ − 4 , begin by finding a perpendicular line I segment to EI through I. Next, bisect that right angle . . . A FOE And you have it. Further, notice that π tan(φ − ) = 4OF. 4
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon K
This circle intersects OB at the point K . Now draw a circle through K with center E ...
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF.
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among K cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF. This circle intersects OB at the point K . Now draw a circle through K with center E ...
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Nearly There.
B Now we use what we know about the relationships among K cos 3α, cos 5α, tan φ, and I tan(φ − π ) to construct the two A 4 FOE lengths we want. Draw a circle with diameter AF. This circle intersects OB at the point K . Now draw a circle through K with center E ...
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon P3
Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon Drumroll . . .
B P 3 This last circle intersects our original diameter at N3 and N5. K I now claim that I A ON = cos 3α N5 FOE N3 3 and
ON5 = − cos 5α.
There is only one thing left to do!
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
P3
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
P3
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish.
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Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish
Daniel Baron Construction of the Heptadecagon Basic Tools and Techniques. Background Constructing Polygons. The Construction. How To Draw a Heptadecagon The Big Finish
Daniel Baron Construction of the Heptadecagon