Download Geometric Constructions for Love and Money

Geometric Constructions
for Love and Money
Kevin Mirus
Madison Area Technical College
Wisconsin Mathematics Council
36th Annual Green Lake Conference
Friday May 7, 2004
[email protected]
http://www.matcmadison.edu/is/as/math/kmirus/
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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…
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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.
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.
Step 2: Crease the paper at
either endpoint of the given
segment. The intersection
of the crease and line are
congruent.
Comments:
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 .
Continued…
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Step 2: Draw an arc centered at P through Q

that intersects PR . Label the intersection
point A.

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.
Step 4: Draw ray QX .
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.
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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.
Problem: Construct an angle congruent to a given angle.
Given: Angle BAC
Step 1: Choose any point D on line l.
Continued…
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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.

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…
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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.
Step 2: Crease the paper on
the other edge of the angle.
The crease and line form a
congruent angle.
Comments:
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.
Comments:
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Step 2: Fold one endpoint
onto the other. This crease
is the perpendicular
bisector.
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?
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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…
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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…
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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 the Quadratix of Hippias.
Given / Step 1: The quadratix of Hippias is a
 y 
curve defined by the equation x  y cot   .
 2
Draw the curve and lay the angle to be trisected
with its vertex at the origin and one edge on the
x axis.
Continued…
Step 2: Mark the point where the angle
intersects the quadratix curve. Construct a line
parallel to the x axis through this point.
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Step 3: Take the segment on the y axis from the Step 4: Draw the line connecting the origin and
origin to the intersection of the horizontal line
the point of intersection between the previous
and divide it into three pieces using a geometric horizontal line and the quadratix. This line
construction. Construct a line parallel to x axis
trisects the original angle.
through the bottom third.
Comments: The quadratix curve can also be constructed “dynamically”…
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Problem: Trisect a given angle using a tomahawk (hatchet).
Comments:
Problem: Divide a given angle into any odd number of equal angles using
hinged tomahawks.
Comments: s
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Problem: Construct a line perpendicular to a given line and containing a
point not on the given line.
Comments:
Problem: Construct a line perpendicular to a given line and containing a
point not on the given line by folding paper.
Fold the line onto itself and form a crease that contains the point.
Comments:
Problem: Construct a line perpendicular to a given line and containing a
point on the given line.
Comments:
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Problem: Construct a line perpendicular to a given line and containing a
point on the given line by folding paper.
Fold the line onto itself and form a crease that contains the point.
Comments: s
Problem: Construct an equilateral triangle.
Comments:
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…
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Problem: Construct a line parallel to a given line through a point not on the
given line.
Comments: s
Problem: Construct a line parallel to a given line through a point not on the
given line.
Comments: s
Problem: Construct a line parallel to a given line through a point not on the
given line using the “draftsman's cheat”.
Comments:
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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.
Comments: This is handy in carpentry for laying out stairs…
Problem: Locate the center of a given circle.
Comments:
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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:
Problem: Construct the circumscribed circle of a given triangle.
Comments:
Problem: Construct the inscribed circle of a given triangle.
Comments:
Problem: Construct the "center of gravity" of a given triangle.
Comments:
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Problem: Construct the two tangents to a given circle through a given point
outside the circle.
Comments:
Problem: Construct a circle given its diameter using a framing square.
Use the fact that an angle inscribed across a diameter is 90.
Comments: This works better than a nail, string, and pencil, I’m told.
Problem: Construct a square.
Comments: You can now construct an octagon, 16-gon, etc.
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Problem: Construct a regular pentagon.
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
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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.
Comments:
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Problem: Construct a triangle congruent to a given triangle.
Comments:
Problem: Construct a regular pentagon using a framing square.
Comments:
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Problem: Construct an approximation to any regular polygon.
Comments:
Problem: Construct a wedge of a 14-gon with seven sticks.
Comments:
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Problem: Construct a Golden Rectangle.
Comments: This is very useful in designing furniture with nice proportions.
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Other Topics: Tangent Circles
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Other Topics: Spirals
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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
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