Lazy Geometry |
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In high school we learn to draw angles and shapes using a straight edge
and a compass. We connect points with the straight edge and marked off
equal segments using the compass. Many seemingly trivial constructions
require several strokes of the pencil and swings of the compass. Grading
and verifying these constructions can be most time consuming. That is why
high school students are never asked to model the space shuttle using only
the straight edge and the compass.
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The compass is not a comfortable device to maneuvre. It is inaccurate for
making arcs of very small radii. On the other hand, a non-trivial amount
of torque is required to make arcs of larger radii. In the true spirit of
laziness, if we minimize the use of the compass, we make cleaner and more
accurate drawings while reducing energy expenditure. So let us pose the
following problem:
Suppose that a swing of the compass requires 5 Calories
of energy, and a stroke of the pencil along a straight edge requires 1 Calorie
of energy. Given a line L and a point A on L,
you are asked to construct a point T such that AT and L
form a 45-degree angle. How many Calories are
required for this construction?
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Typically, the first step is to construct a right angle. Then in the
second step, the right angle is bisected, thus forming the desired
45-degree angle. The picture on the left shows this construction with
minimal optimization. The steps are:
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Using the compass, construct a circle centered at A,
intersecting line L at P and Q
(5 Calories).
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Increase the radius of the compass.
Construct two circles,
one centered at P and one centered at Q,
both of the same larger radius
(10 Calories).
These two circles intersect at R.
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Using the straight edge, construct line RA
(1 Calorie).
Line RA is perpendicular to line L,
and intersects the first circle at S.
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Using the compass, construct a circle centered at S
of the same radius as the two circles in step 2
(5 Calories).
(This radius is equal to the length of RQ).
This fourth circle and the circle centered at Q
intersect at T.
Lines AT and L form a 45-degree angle.
Total: 4 circles and 1 line requiring 21 Calories.
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We can optimize this construction a little bit. After the construction
of S in step 3, we already have the 45-degree angle formed by lines
PS and L. If we shift our construction, we save 5 Calories.
The steps are:
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Choose an arbitray point P on line L.
Using the compass, construct a circle centered at P
of radius PA,
intersecting line L at A and B
(5 Calories).
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Increase the radius of the compass.
Construct two circles,
one centered at A and one centered at B,
both of the same larger radius
(10 Calories).
These two circles intersect at R.
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Using the straight edge, connect R and A
(1 Calorie).
Line RA intersects the first circle at T.
Lines AT and L form a 45-degree angle.
Total: 3 circles and 1 line requiring 16 Calories.
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The first method involves the construction of the perpedicular
bisector followed by that of an angle bisector. The second method
circumvents the construction of the angle bisector by shifting
the right angle. However, there is a lazier way to construct
a right angle. The steps are:
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Choose an arbitrary point P not on line L.
Using the straight edge, construct line PA
(1 Calorie).
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Using the compass, construct a circle centered at P
of radius PA
(5 Calories).
This circle intersects line L at B
and line PA at C.
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Using the straight edge, construct line BC
(1 Calorie).
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Using the compass, construct a circle centered at B
of radius BA
(5 Calories).
This circle intersects line BC at T.
Lines AT and L form a 45-degree angle.
Total: 2 circles and 2 lines requiring 12 Calories.
By applying the fact that the hypotenuse of a right triangle
equals the diameter of its circumscribing circle, we save
four Calories.
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Homework: What is the laziest way to construct a regular pentagon
using a straight edge, a compass and a piece of paper?
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Answer
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