A Brief Look into Geometric Constructions Focusing on Basic Constructions and the Impossible Constructions Bethany Das, Emily Simenc, and Zach Tocchi . Basic Constructions Impossible Constructions Bisecting a Segment/Creating a Perpendicular to a Segment Abstract Constructing a 7-sided Regular Polygon When it comes to constructing regular polygons, one would think that any shape can be created with a compass and a straightedge inside of a circle. We showed this to be true with a regular pentagon (lower left of the poster) and in activities, we showed that it is possible to construct a regular hexagon, but it is actually only possible to construct polygons of certain properties. For instance, one can only construct a polygon if it has powers of two as the numbers of sides (4, 8, 16, etc.) as well as Fermat’s primes (3, 5, 17…) or any product of numbers from the two sets. This is because when 360 is divided by those numbers, the fraction is rational. And yes, you can really construct a 65,537 sided polygon. Better sharpen lots of pencils though! To make a construction, one may only use a compass and a straightedge with no markings. Throughout mathematics history, it has been a challenge for people to create these constructions with only the tools given. Through much trial and error, we now can easily create various line, angle, circle, inscribed and polygon constructions. However, even after conquering all these constructions, there are still some that have not been done. It is not that we, or others, are not talented or smart enough to do them, it is that they are impossible to construct by hand. The purpose of our project is to explore constructions that are fairly easy to make, to constructions that are more complex as well as impossible constructions. Through the use of discovery-based activities, we demonstrate to students why these constructions are impossible. Construct segment, ,using a straightedge. Estimate what the midpoint of would be and label it M. M should be about halfway between points A and B and on . Now, with compass, construct a circle with center, A, and that goes through any point on . Now, with compass at the exact same setting, construct a circle with center, B, and that goes through any point on . Circles A and B should intersect at two points. Label these points C and D. Construct segment . Trisecting an Angle Constructing an Equilateral Triangle History of Constructions We know that by using circles, we can trisect a segment and bisect an angle. It would be natural to think that by using a combination of these techniques, we can then trisect an angle. Right? Wrong! The problem with any curve used to trisect an angle, is that even though you can construct an infinite number of points on the curve using a straightedge and compass, you cannot construct every single point. No matter how close the points are spaced, there is always going to be very small gaps between them. So, any curve drawn, will always be approximate, not exact. So, it is impossible to trisect an angle since every angle relates to the solution of a certain cubic equation which is impossible to do with only a compass and a straightedge. Construct segment, , using straightedge. With compass, construct a circle with center, A, and that goes through B. Now, with compass, construct a circle with center, B, and that goes through A. Circles A and B should intersect at two points. Label these points C and D. Construct segments and using straightedge. Why only straightedge and compass? Greeks could not do arithmetic. Only had whole numbers. No zero, no negatives, no decimals. When faced with a problem such as finding the midpoint of a segment, they could not do the obvious - measure it and divide by two. Had to find other ways, which lead to the constructions using straightedge and compass. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute using arithmetic. Roots in Greek mathematics Oenopides of Chios (fl. 450 BC) was one of the more celebrated mathematicians of the era Credited with being the first to construct a perpendicular and to construct an angle equal to a given angle, but it is more likely that he was the first to do this using only straightedge and compass. Euclid’s geometry is the geometry of straight lines and circles, and the figures that can be derived from them. Discussed this geometry and constructions in his Elements. Complex Constructions Constructing an Inscribed Pentagon Squaring a Circle Squaring the circle is a problem posed by ancient geometers as a challenge. They were told to construct a square with the same area as a given circle by using a finite number of steps with a compass and straightedge. The Lindermann-Weierstrass theorem proved that pi is an irrational number which means it is not the root of any polynomial with rational coefficients. Decades before, mathematicians said that if pi were an irrational number, then the construction would be impossible. Which leads to the case that squaring the circle can only be approximated using rational numbers arbitrarily close to pi. Using your compass, construct a circle with center at point O. Also draw a diameter through the center of the circle. Label your diameter with points A and B. Next, create a perpendicular bisector to the diameter of the circle. This bisector should also be a diameter. Label this diameter with points C and D. Draw an arc using as the length of the arc. Make sure this arc crosses the circle at two points E and F. Draw . Label the point where crosses point G. Draw . Draw an arc with as the radius centered at G. This will cross your newly created segment at one point. Call that point H. Draw an arc centered at C with radius . Make sure it crosses circle O at two points! Label these points J and K. Also draw . This is the first side of your pentagon! Draw an arc centered at point J using as its radius. This arc must cross your circle at one point, call it L and construct . Connect point L with point D creating . Repeat steps 6 and 7 to create a point M opposite L. Also connect this point to point D. You should now have segments and . Drawing Parallel Lines 1. Start by drawing the line segment and a point R off the line. 2. Draw a transverse line through R and across the line at an angle, forming the point J where it intersects the line . The exact angle is not important. 3. With the compass width set to about half the distance between R and J, place the point on J, and draw an arc across both lines. 4. Without adjusting the compass width, move the compass to R and draw a similar arc to the arc in the previous step. 5. Set compass width to the distance where the lower arc crosses the two lines. 6. Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S. 7. Draw a straight line through points R and S. This line is parallel to line . Compass Can be opened arbitrarily wide Contains no markings Assumed to collapse when lifted from the page May not be directly used to transfer distances Straightedge Assumed to be of infinite length Only one edge Can only be used to draw a line segment between two points or to extend an existing line Course Information: 3450:441-001 Concepts in Geometry Dr. Antonio Quesada R S P Q More Constructions and Activities are Available as Printouts.