Tag Archives: compass and straightedge

The Power of Construction

Eighth grade Geometry was one of my favorite classes. I thoroughly enjoyed the material, as it was unlike anything I had seen in my math education so far. Personally, I felt that Geometry helped combat the classic refrain of nearly every elementary school child of “When am I going to use this”. Geometry had the unique ability to take all of the abstract ideas taught thus far and ground them in the physical world. It was also the first time we were required to write a logical proof, which has been such an important skill to have and understand. However, my favorite part of the class was the constructions.

I clearly remember being excited about buying my very first compass. I knew that with my compass and stainless steel ruler I used as a straightedge, I was ready for anything Geometry could throw at me. In class, we constructed all of the basics: perpendicular bisectors, bisected angles, various regular polygons, and so on. The pentagram construction was my favorite. That knowledge of that construction, combined with chalk, some string, a yardstick and the driveway made for a very interesting conversation with my parents. Construction is a great hands-on approach to math but is rarely seen past the context of 8th grade geometry.

After 8th grade, I never gave it any thought. It wasn’t until my History of Math course that I realized how much power straightedge-and-compass construction really had.

So let us go back to the basics. A straightedge is a ruler without any graduation. It can be used in straightedge-compass constructions to connect two points on a given plane, and extend lines on a given plane. The compass is used to draw circles (that is, a set of all points equidistant from another point), and be able to ‘measure’ a given line segment, and construct that same length elsewhere. So what can you do with this?

As it turns out, quite a bit! For starters, you can do basic arithmetic. Adding just becomes combining two line segments on the same line, and subtraction is the reverse. Multiplication has a geometric representation as similar triangles. If you have a triangle whose base is 1 unit and a hypotenuse of length a, if you draw a similar triangle whose base is length b, the resulting hypotenuse is length ab.

Trisecting a segment. Image: Goldencako, via Wikimedia Commons.

I find this incredible! What took my math educators 8 years to get around to, the Ancient Greeks did right off the bat. They made mathematics immediately tangible and constructible, instead of relying on the esoteric notion of numbers and Algebra. But construction doesn’t stop there. As I’ve mentioned you can create perpendicular lines, create regular polygons, bisect angeles, bisect segments, trisect segments, and more. Given a unit length, you can even construct whole-number measurements of that length, and even some irrationals like square roots. Take this example:

Construction of a square. Image: Aldoaldoz, via Wikimedia Commons.

To construct a regular unit square (all sizes equal to a given unit of measure and all angles are right angles), it’s a simple matter of constructing perpendicular bisectors of segments and measuring with a compass. But can we construct a square that is exactly twice as much area? If the area of a unit square is 11 = 1then the area of a doubled square must be 2. The sides of the square must then be 2 = s2 ; s = 2. To the uninitiated, this might seem like an impossible task! How on earth using a straightedge and a compass can you construct an irrational number? Euclid, however, found a way. Create two lines that are perpendicular to each other. Use the compass to measure out a unit length along each of the two lines, starting from the intersection. This has given you two sides of a right triangle, each with a length of 1 unit. If you connect them an form a hypotenuse, a2+b2= c2; 1 + 1 = c2; c = (1+1) = 2. With this new length as a measurement for your compass, a square with side length 2 is entirely possible.

Euclid did have one problem them though, and that was cube roots. We have shown that doubling the square is possible, but what about doubling the cube? The construction is analogous to the double the square, just with an added dimension. Therefore, if a cube has a volume of 1 cube unit, it’s double should have an volume of 2 cubed units. V = s s s = s3; s = 3V

No matter how hard Euclid tried, he could not construct a cube root. His limitations didn’t stop there. Most famously, he was unable to construct a square the with the area of a circle. This is known as “Squaring the circle.” Again, if we have a circle with the radius of 1 unit, it will have an area of A = r2 =. So to make a square have an area of , we simply have to construct a side with length . Square roots are no problem, it’s just the hypotenuse of a right triangle with sides that sum to . But…how do we sum to with constructions? It turns out, it is impossible. The number is not a “constructible number, ” as they are known, but a “transcendental number.” This wasn’t proved until 1883 by Ferdinand von Lindemann.

While I would never give up the power of algebra and the tools it provides, Euclidean geometry holds a special place in my heart for its sheer physicality. The ability to construct basic arithmetic, regular polygons, and even the odd irrational number grounds math in a way that I think is delayed for far too long in standard Western education. But at least they get around to it.

I end, as always, with wise words from Randall Munroe.

Image: xkcd by Randall Munroe.





The Forgotten Mathematician

Pierre Wantzel was born in 1814 on the 5th of June in Paris. His father was a professor of mathematics at École speciale du Commerce after serving in the army. Due to this Pierre started his life with a natural love for mathematics. He attended school at his home town of Ecouen where he demonstrated this love. When he was only 9 years old his teachers would turn to him for help when judging the difficulty of problems. His love and skills for mathematics was realized by his parents when they sent him to École des Arts et Métiers de Châlons. He was surprisingly 12 years old he went there, and this was far younger than most.   His teacher was the well-known Étienne Bobillier, a mathematician known for his works on polar curves and algebraic surfaces. This helped kinder his mathematical skill, but it did not last long because in 1827 the school was reformatted. This was because France itself was facing revolts and other political issues. The school was reformed to become less academic, and this caused Pierre to take his studies elsewhere.

In 1828 he traveled to the Collège Charlemagne to continue his studies and receive language coaching. He later married the daughter of language coach, but before this he accomplished many feats of genius including editing a second edition book by Reynauld, Treatise on arithmetic, at only 15 in 1829. This book featured a method for finding square roots that was never proved. He proved the method, and in doing so he received the first prize for dissertation from his college. Later on he took the entrance exam to École Polytechnique and the science section for École Normale. He placed first in both of these, something never before achieved. Furthering his education he traveled to Ponts et Chaussées, an engineering school, but did not stay long. He remained there for a year until 1835 where he journeyed to the Ardennes. Following a similar pattern he later traveled to Berry after only a year at the Ardennes. After studying engineering he decided that teaching mathematics was his true dedication. In order to achieve this he took a leave from his occupation, and went to become a lecturer for a school from his past, École Polytechnique. He later became a professor of applied mathematics at École des Ponts et Chaussées but not before becoming an engineer in 1841. Continuing with his true interests he began teaching classes on not only mathematics but physics as well. He continued his educational career becoming the entrance exam examiner In 1843. He was not confined by his university, however, as he traveled around Paris to many schools teaching there too.

The tools of Pierre Wantzel. Image: Mcgill via Wikimedia Commons.

Pierre achieved fame when he published what would become his most important works. These were on the subject on radicals, and solving equations and they were dubbed as some of the most famous problems of the time. Publishing them in Liouville’s Journal he was the first to prove that it was impossible to duplicate a cube and trisect an angle with a ruler and compass. Gauss had originally stated that it was impossible but offered no proof. This is what Pierre accomplished in his 1837 paper where he traces the solution back to cube roots, something that proves impossible to do with those tools. This was built off of the work of others, yet it still went beyond what had been previously done. Continuing his works Pierre delved into equations, and from this he created new proofs of algebraic equations deemed impossible. These were solved not by providing a solution but proving they were impossible to solve. He revised a proof of Abel’s theorem in 1845, stating that it was impossible to solve any equations where the exponent n is greater than 5. He also added details to many vague solutions on the subject these solutions were proposed by famous mathematicians such as Ruffini. Pierre published over 20 works throughout the course of his life a few of these branch out into the field of physics, specifically dealing with extreme pressure differences.

Pierre was a strong man who focused on his work so much that he sacrificed his sleep and meals to do so. Pierre Wantzel did not live out his life fully as he overworked himself. He relied on coffee and opium to continue his lifestyle, and this ultimately resulted in his demise. In 1848 at the age of 33 he died and the world lost a great mind. Overall his works were very important yet were not remembered as well as others. This is commonly attributed to the classical nature of the problem he is famous for. Several other mathematicians mentioned the problem yet they have given no proof. Max Simon’s work from 1906 does mention Pierre’s, but it was published as a supplement to another work rather than as its own. Another reason is his early death. Due to his potential yet little time to achieve true greatness he is less known. Sadly he was not elected as a member of the Académie des Sciences. His achievements were great and if he had lived longer he would have achieved much more.