Sketchpad (and other) Constructions Kevin Mirus Madison Area Technical College Madison East High School Math Week Monday, May 16, 2005 kmirus@matcmadison.edu http://www.matcmadison.edu/is/as/math/kmirus/ Page 1 of 20 Some types of geometric construction (Ref. 1): 1. Euclidian (Ruler and Compass) constructions use a ruler with no markings and a compass which can only be used to draw a circle with a given center point through another given point. Angles cannot be trisected under these restrictions. 2. Mohr – Mascheroni (Compass Only) constructions use a compass alone; anything that can be constructed with a ruler and compass can be constructed with a compass alone (!). 3. Ruler constructions use a ruler alone; anything that can be constructed with a ruler and compass can also be constructed with just a ruler and four given vertices of a trapezoid (!). 4. Ruler and Dividers constructions use a ruler and a divider to carry distance. This allows some constructions that a ruler and compass don’t. 5. Poncelet-Steiner and Double Ruler constructions use either a ruler and a circle with a known center or a ruler with two parallel edges. All ruler and compass constructions can be constructed under these circumstances. 6. Ruler and Rusty Compass constructions use a ruler and a compass of fixed opening; anything that can be constructed with a ruler and compass can be constructed with a compass alone (!). 7. Stick constructions use sticks of equal length that can be laid on top of each other. You can do more constructions with sticks than with a ruler and compass! 8. Marked Ruler constructions use a ruler with two marks on its edge. You can do more constructions with a marked ruler than with a ruler and compass. 9. Paperfolding constructions use folds in paper. Any constructions of plane Euclidian geometry can be executed by means of creases. 10. Constructions using other instruments like a “tomahawk”, carpenter’s squares, strings, etc. allow for constructions not possible using a ruler and compass. 11. Approximate constructions are useful when close is good enough… Page 2 of 20 Problem: Construct a line segment congruent to a given segment. Step 2: Use a compass to measure the distance AB, then draw an arc centered at C with radius AB that intersects l at point D. Comments: This is not a strict Euclidian construction, since the compass was used to transfer a distance. This construction can also be used to construct the sum and difference of two given segments. This construction can also be used to construct an equilateral triangle (see below), or to copy any triangle, which then allows you to copy any polygon (because any polygon can be broken up into triangles). Given: Line segment AB Step 1: Choose any point C on line l. Problem: Construct a line segment congruent to a given segment by folding paper. Given: A line segment. Step 1: Fold the given line segment onto the other line. Comments: Problem: Construct an equilateral triangle. Comments: Page 3 of 20 Step 2: Crease the paper at either endpoint of the given segment. The intersection of the crease and line are congruent. Problem: Construct a circle whose radius is equal to the length of a given segment. Given: Line segment PQ , point R Step 1: Draw ray PR . Step 2: Draw an arc centered at P through Q that intersects PR . Label the intersection point A. Step 4: Draw ray QX . Step 3: Draw arcs centered at points Q and A through point P. Label their intersection X. Step 5: Draw an arc centered at P through R that intersects PQ . Label the intersection point B. Page 4 of 20 Step 6: Draw arcs centered at points R and B through point P. Label their intersection Y. Step 7: Draw ray RY . Label the intersection of RY and QX point S. Step 7: Draw the circle centered at R through S. RS = PQ, since PQRS is a parallelogram. Thus, any radius of this circle is equal to PQ. Comments: This is a strict Euclidian construction for congruent segments. Its importance is to show that a Euclidian compass is equivalent to a modern compass (i.e., it's okay to use a compass to remember distances). Ref. 1. Page 5 of 20 Problem: Construct an angle congruent to a given angle. Given: Angle BAC Step 1: Choose any point D on line l. Step 2: Draw an arc of any radius centered at A. Label the points of intersection P and Q. Create the same radius arc centered at D. Label the point of intersection E. Step 3: Draw an arc centered at P through Q. Draw an arc centered at E with the same radius. Label the intersection point F. Page 6 of 20 Step 4: Draw ray DF . BEDF is congruent to BAC Comments: This construction can also be used to construct the sum and difference of angles. A good exercise is to construct a 105 angle using ruler and compass… Problem: Construct an angle congruent to a given angle by folding paper. Given: An angle. Step 1: Fold the given angle onto the other line. Comments: Page 7 of 20 Step 2: Crease the paper on the other edge of the angle. The crease and line form a congruent angle. Problem: Construct the perpendicular bisector of a given line segment. Step 1: Draw arcs centered Step 2: Draw segment PQ . on A and B that have the Label its intersection with same radius greater than the AB as point M. distance to the midpoint. Label the intersections P and Q. Comments: Constructing the perpendicular bisector gives you the midpoint for free! Given: Line segment AB Problem: Construct a perpendicular bisector by folding paper. Given: A segment. Step 1: Create the crease between the endpoints. Step 2: Fold one endpoint onto the other. This crease is the perpendicular bisector. Comments: Problem: Construct a line perpendicular to a given line and through a given point. Starting with line segment AB and point C not on the line, strike an arc centered at C so that the arc intersects AB at two points (E and F). Construct the perpendicular bisector between EF, and it will pass through point C. Comments: Page 8 of 20 Problem: Construct a line perpendicular to a given line and through a given point by folding paper. Given: A segment and a point. Step 1: Crease the paper so that the line segment folds onto itself and the crease contains the given point. Comments: Problem: Construct the bisector of a given angle. Given: Angle BAC Step 1: Draw an arc of any radius centered at A. Label the points of intersection P and Q. Step 2: Draw arcs of equal radius from P and Q. The line from A through their intersection bisects BAC . Comments: Problem: Construct the bisector of a given angle by folding paper. Given: An angle. Step 1: Fold one edge of the angle onto the other edge. Comments: Cool, huh? Page 9 of 20 Step 2: The crease bisects the angle. Problem: Trisect a given angle using Pappus' Theorem / Method of Archimedes. Given: Angle ABC Steps: Construct perpendiculars and parallels through A. Draw an arc centered at A that passes through B in the interior of the circle. Draw ray BS such that MS = RM = AB. Comments: This is a marked ruler construction… Problem: Trisect a given angle using Archimedes' method. Given: Angle ABC Steps: Draw a circle centered at B that passes through A and C. Draw line AQ such that PQ = BC = AB = PB. Comments: This is another marked ruler construction… Page 10 of 20 Problem: Trisect a given angle by folding paper. Given: Angle PQR . Call PQ line l, and QR line m. Step 2: Construct the perpendicular to line m through point M by folding line m onto itself such that the crease contains point M. Call the crease that is made line b. Step 1: Locate the midpoint M of PQ by folding line l onto itself so that point P lands on top of point Q. Call the crease that is made line a. Step 3: Construct the perpendicular to line b through point M by folding line b onto itself such that the crease contains point M. Call the crease that is made line c. Notice that line c is parallel to line m. Continued… Page 11 of 20 Step 4: Fold the paper so that point P lands on Step 5: Fold the paper so that the crease line b and point Q lands on line c. This is tricky contains points Q and Q’. Call the crease line e. and will take some sliding around to get right. Line e trisects PQR . On the diagram above, point P landed on top of line b at point P’, while point Q landed on line c at point Q’. Call the crease that is made line d. Comments: You can do more with paperfolding than with a ruler and compass! Problem: Trisect a given angle using a tomahawk (hatchet). Comments: Page 12 of 20 Problem: Construct an equilateral triangle by folding paper. Make a square, place a crease up the middle, then fold a vertex of the square onto the middle crease. Comments: Problem: Construct an equilateral using the vesica pisces. Comments: Things are always a little more interesting with subversion and religion mixed in… Problem: Construct a line parallel to a given line through a point not on the given line. Given line AB, and point C not on the line, draw line AC. Then, strike a random-length arc centered at A that has intersection points D and E. Strike the same radius arc centered at C that has intersection point F. Use a compass to measure the distance from D to E, then mark off that same distance from point F to locate point G. The line through CG is parallel to AB. Comments: Page 13 of 20 Problem: Construct a line parallel to a given line through a point not on the given line by paper folding. Construct two perpendiculars… Comments: Problem: Partition a given line segment into a given number of congruent segments. To divide the original segment AB into 5 parts, draw a ray at any random angle, then mark it off in 5 equal parts until you get to point C. Draw a line through BC, then construct parallels to BC through each point on AC. The intersection points on AB will divide it into 5 equal parts. Comments: This is handy in carpentry for laying out stairs. Page 14 of 20 Problem: Locate the center of a given circle. Comments: Problem: Construct the tangent line to a circle at a given point on the circle. Draw a radius to the point, then construct a perpendicular to the radius. Comments: Page 15 of 20 Problem: Construct the two tangents to a given circle through a given point outside the circle. Comments: Problem: Construct a regular pentagon. Step 1 Step 2 Page 16 of 20 Step 3 Step 4 Step 5 Step 6 Step 7 Comments: s Problem: Construct a regular hexagon. Comments: This forms the basis for many a church window. Extending the pattern makes a pretty tiling. Page 17 of 20 Problem: Construct a regular hexagon using two vesica pisces. Comments: Problem: Construct an approximation to any regular polygon. Comments: Problem: Construct a wedge of a 14-gon with seven sticks. Comments: Page 18 of 20 Problem: Construct a Golden Rectangle. Start with square ABCD, then find the midpoint M of side CD. Strike an arc centered at M that starts at A and stops when it crosses ray CD (at point X). Construct point Y perpendicularly above point X, and the new rectangle BCXY is a golden rectangle. Comments: This is very useful in designing furniture with nice proportions. The Golden Ratio, as embodied in the Golden Rectangle, is the most pleasing rectangle and proportion to the human eye. The ratio of the length to the width of a Golden Rectangle 1 5 :1 1.618 :1 8 : 5 . Use it when designing furniture or buildings. is 2 Page 19 of 20 References 1. Isaacs, I. Martin, Geometry for College Students, Brooks/Cole Thomson Learning, 0-534-35179-4, 2001. 2. Posamentier, Alfred S. and William Wernick, Advanced Geometric Constructions, Dale Seymour Publications, 0-86651-429-5, 1988. 3. Martin, George E., Geometric Constructions, Springer, 0-387-98276-0, 1998. 4. Sykes, Mabel, Source Book of Problems for Geometry, Dale Seymour Publications, 0-86651-795-2, 1912. 5. "Four Ways to Construct a Golden Rectangle," Fine Woodworking, Taunton Press, February 2004, No. 168, p. 50. 6. Lundy, Miranda, Sacred Geometry, Wooden Books, 0-965-20578-9, 1998. 7. Bedford, John R., Graphic Engineering Geometry, Gulf Publishing Company, 087201-325-1, 1974. 8. "Details on Involute Gear Profiles," How Stuff Works web site (has a great animation of gears meshing), http://science.howstuffworks.com/gear7.htm 9. "Origami and Geometric Constructions," http://www.merrimack.edu/~thull/omfiles/geoconst.html 10. Machinery's Handbook (any edition) 11. "Logarithmic Spiral," mathworld website, http://mathworld.wolfram.com/LogarithmicSpiral.html 12. Wingeom for Windows, website with freeware for geometric constructions (used to create all diagrams in this packet) http://math.exeter.edu/rparris/wingeom.html 13. “Tangent Circles,” a web site: http://whistleralley.com/tangents/tangents.htm 14. “Constructing the Heptadecagon,” a web site: http://www.mathpages.com/home/kmath487.htm Page 20 of 20