"She [yesterday's gardener] had no idea that the garden abounded with equiangular spirals. They were in the seed-heads of the daisies and various flowers."
So Pappas writes about an equilateral spiral, also known as a logarithmic spiral. Equilateral spirals are so-called because at each point on the spiral, the angle formed by segment joining that point to the center of the spiral and the tangent (in the Calculus sense) at that point is constant. Circles also have that property -- the constant angle equals 90 degrees. Equiangular spirals occur when the constant angle is not 90.
Equiangular spirals are self-similar in that there exists a dilation mapping the spiral to itself. The center of the dilation is, of course, the center of the spiral. The magnitude k of this dilation depends only on the constant angle theta -- k = e^(2pi/tan theta). If k is known, then we can solve this for theta to obtain theta = arctan(2pi/ln k) -- and there's the "logarithm" in the name. It turns out that a dilation of other scale factors is actually equivalent to a rotation of the spiral!
Lesson 15-5 of the U of Chicago text is on "Angles Formed by Chords or Secants." There is one vocabulary term as well as two theorems to learn.
The vocabulary word to learn is secant. The U of Chicago defines a secant as a line that intersects a circle in two points. This is in contrast with a tangent, a line that intersects the circle in one point.
At this point, I often wonder why we have tangent and secant lines as well as tangent and secant functions in trig. Well, here's an old (nearly 20 years!) Dr. Math post with the explanation:
Now, the tangent and the secant trigonometric functions are related to the tangent and secant of a circle in the following way. Consider a UNIT circle centered at point O, and a point Q outside the unit circle. Construct a line tangent to the circle from point Q and call the intersection of the tangent line and the circle point P. Also construct a secant line that goes through the center O of the circle from point Q. The line segment OQ will intersect the circle at some point A. Next draw a line segment from the center O to point P. You should now have a right triangle OPQ. A little thought will reveal that the length of line segment QP on the tangent line is nothing more but the tangent (trig function) of angle POQ (or POA, same thing). Also, the length of the line segment QO on the secant line is, not surprisingly, the secant (trig function) of angle POA.
And now let's look at the theorems:
The measure of an angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its vertical angle.
Prove: Angle CEB = (Arc AD + Arc BC)/2
2. Angle C = Arc AD/2, 2. Inscribed Angle Theorem
Angle A = Arc BC/2
3. Angle CEB = Angle C + Angle A 3. Exterior Angle Theorem
4. Angle CEB = Arc AD/2 + Arc BC/2 4. Substitution
The measure of an angle formed by two secants intersecting outside the circle is half the difference of the arcs intercepted by it.
Prove: Angle E = (Arc AC - Arc BD)/2
2. Angle ADC = Arc AC/2, 2. Inscribed Angle Theorem
Angle A = Arc BD/2
3. Angle A + Angle E = Angle ADC 3, Exterior Angle Theorem
4. Angle E = Angle ADC - Angle A 4. Subtraction Property of Equality
5. Angle E = Arc AC/2 - Arc BD/2 5. Substitution
In the end, I must admit that of all the theorems in the text, I have trouble recalling circle theorems the most.
I decided to include another Exploration Question as a bonus:
The sides of an inscribed pentagon ABCDE are extended to form a pentagram, or five-pointed star.
a. What is the sum of the measures of angles, F, G, H, I, and J, if the pentagon is regular?
Notice that each angle satisfies the Angle-Secant Theorem. So Angle F is half the difference between CE (which is two-fifths of the circle, Arc CD + Arc DE = Arc CE = 144) and AB (which is one-fifth of the circle, Arc AB = 72). So Angle F = (144 - 72)/2 = 36 degrees. All five angles are measured the same way, so their sum is 36(5) = 180 degrees.
b. What is the largest and smallest this sum can be if the inscribed polygon is not regular.
Well, let's write out the Angle-Secant Theorem in full:
Angle F + G + H + I + J
= Arc (CD + DE - AB + DE + EA - BC + EA + AB - BC + AB + BC - DE + BC + CD - EA)/2
= Arc (CD + DE + EA + AB + BC)/2
= (360)/2 (since the five arcs comprise the entire circle)
So the largest and smallest this sum can be is 180. The sum of the five angles is a constant.