Last week
while I was waiting to pick up my kids from their school carpool line, I
noticed that all the windows in the church building had a certain design. Despite being different widths and heights, each
window was symmetrical, where both left and right sides curved up to a point,
the outermost curves continued on to meet at another point, and there was a circle
in the top middle that was just the right size to perfectly touch all the other
curves in one spot each.
I began to
wonder about that middle circle. How did
the architect find the correct size of that circle so that it was tangential to
the other curves? Was it by trial and
error? Or was there a formula for
it? Could a general formula be found for
it?
To simplify
things, let’s assume that all the curves were made by circles (and not by some
other conic section like a parabola, ellipse, or hyperbola). The four different curves can then be
constructed by using four congruent circles as guidelines.
Each window
can then be defined by two variables: the radius r of the four congruent
circles, and the distance w between the two nonsymmetrical arcs (or half the
width of the whole window). Since each
window design is symmetrical, the center of the middle circle with the unknown
radius x will lie on the vertical line of symmetry. We can now construct two right triangles with
the following sides as pictured below:
The left-side
triangle has a hypotenuse of r – x and legs r – w and y, which makes the
Pythagorean equation y2 + (r – w)2 = (r – x)2
or y2 = (r – x)2 – (r – w)2. The right-side triangle has a hypotenuse of r
+ x and legs r and y, which makes the Pythagorean equation y2 + r2
= (r + x)2 or y2 = (r + x)2 – r2. Therefore,
|
y2 = (r – x)2 – (r – w)2 = (r + x)2
– r2
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(Pythagorean’s Theorem)
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(r2–2rx+x2) – (r2–2rw+w2)
= (r2+2rx+x2) – r2
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(expand)
|
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-2rx + x2 + 2rw – w2 = 2rx + x2
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(simplify)
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-2rx + 2rw – w2 = 2rx
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(subtract x2)
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2rw – w2 = 4rx
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(add 2rx)
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x = (2rw – w^2)/4r
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(divide 4r)
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The radius x
of the middle circle is therefore
where r is
the radius of one of the larger four congruent circles and w is half the width
of the whole window. So if the radius of
one of the larger circles is 3 feet, and half the width of the window is 2
feet, then the radius of the middle circle would be x = (2rw – w^2)/4r
= (2(3)2 – 2^2)/4(3) = 8/12 feet
or 8 inches. Or if the radius of
one of the larger circles is 8 feet, and half the width of the window is 4
feet, then the radius of the middle circle would be x = (2rw – w^2)/4r
= (2(8)4 – 4^2)/4(8) = 48/32 = 3/2
feet or 18 inches.
As a side
note, calculus lovers should recognize that if r remains constant, the maximum
radius x of the middle circle occurs when w = r. This is because dx/dw =
1/4r(2r – 2w), and setting this to zero gives the
solution w = r. Geometrically, w = r
occurs when the top of the window does not come to a point but is a smooth
semicircle.
There are
probably other church window designs that have similar properties. For this particular design, the size of the middle
circle can be calculated exactly.
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