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Straight-Edge and Compass: Constructing the Heptadecagon

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Straight-Edge and Compass: Constructing the Heptadecagon 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
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