Brief Outline of Types of Construction

Sketchpad (and other) Constructions

Kevin Mirus Madison Area Technical College

Madison East High School Math Week Monday, May 16, 2005

[email protected] http://www.matcmadison.edu/is/as/math/kmirus/

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…

of 20 Problem: Construct a line segment congruent to a given segment.

Given: Line segment AB Step 1: Choose any point C Step 2: Use a compass to on line l. 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).

Problem: Construct a line segment congruent to a given segment by folding paper.

Given: A line segment. Step 1: Fold the given line Step 2: Crease the paper at segment onto the other line. either endpoint of the given segment. The intersection of the crease and line are congruent.

Comments:

Problem: Construct an equilateral triangle.

Comments:

of 20 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 Step 3: Draw arcs centered at points Q and A  through point P. Label their intersection X. that intersects PR . Label the intersection point A.

 Step 5: Draw an arc centered at P through R Step 4: Draw ray QX . that intersects PQ . Label the intersection point B.

of 20 Step 6: Draw arcs centered at points R and B Step 7: Draw ray  . Label the intersection through point P. Label their intersection Y. RY   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.

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 Step 3: Draw an arc centered at P through at A. Label the points of intersection P and Q. Draw an arc centered at E with the Q. Create the same radius arc centered at same radius. Label the intersection point F. D. Label the point of intersection E.

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 Step 2: Crease the paper on angle onto the other line. the other edge of the angle. The crease and line form a congruent angle.

Comments:

of 20 Problem: Construct the perpendicular bisector of a given line segment.

Given: Line segment AB 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 as point M. distance to the midpoint. AB Label the intersections P and Q. Comments: Constructing the perpendicular bisector gives you the midpoint for free!

Problem: Construct a perpendicular bisector by folding paper.

Given: A segment. Step 1: Create the crease Step 2: Fold one endpoint between the endpoints. 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:

of 20 Problem: Construct a line perpendicular to a given line and through a given point by folding paper.

Given: A segment and a Step 1: Crease the paper so that the line segment folds point. 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 Step 2: The crease bisects the angle onto the other the angle. edge.

Comments: Cool, huh?

of 20 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…

of 20 Problem: Trisect a given angle by folding paper.

   Given: Angle PQR . Call PQ line l, and QR Step 1: Locate the midpoint M of PQ by line m. folding line l onto itself so that point P lands on top of point Q. Call the crease that is made line a.

Step 2: Construct the perpendicular to line m Step 3: Construct the perpendicular to line b through point M by folding line m onto itself through point M by folding line b onto itself such that the crease contains point M. Call the such that the crease contains point M. Call the crease that is made line b. crease that is made line c. Notice that line c is parallel to line m.

Continued…

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:

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.

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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:

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.

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:

of 20 Problem: Construct the two tangents to a given circle through a given point outside the circle.

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Problem: Construct a regular pentagon.

Step 1 Step 2

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.

Problem: Construct a regular hexagon using two vesica pisces.

of 20 Problem: Construct an approximation to any regular polygon.

Comments:

Problem: Construct a wedge of a 14-gon with seven sticks.

Comments:

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 is :1换 1.618:1 8: 5 . Use it when designing furniture or buildings. 2

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, 0- 87201-325-1, 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