Astroids

Imagine a ladder falls down so that the top of the ladder slides down the wall and the base of the ladder slides along the ground away from the wall.  If you were to record the falling ladder from the side and overlay each frame onto a single picture, the ladders from the different frames would form a curve.

Falling Ladder

At first glance it may appear that this curve is circular.  After all, the top of the curve is measured when the ladder is leaning straight up against the wall, and the bottom of the curve is measured when the ladder is lying flat on the ground, which would appear to show a radius that is the same length as the ladder.  However, if we compare our falling ladder curve with a curve from a circle, we can see that the circular curve is slightly steeper.

Falling Ladder Curve vs. Circular Curve

Obviously, some more calculations will be needed to find an equation for this curve.  We can start by calling the wall the y-axis and the ground the x-axis.  Then if we call the length of the ladder w, and the angle between the ladder and the ground θ, then the distance from the top of the ladder to the origin (along the wall) is w sin θ and the distance from the bottom of the ladder to the origin (along the ground) is w cos θ.  We can also label a point (x, y) somewhere on the ladder.

There are then two similar triangles that we can set in proportion to each other: the large triangle with sides w sin θ and w cos θ, the smaller triangle with sides y and w cos θ – x.  Therefore:

y / w cos θ – x = w sin θ / w cos θ	(similar triangles)
yw cos θ = w sin θ (w cos θ – x)	(cross multiply)
yw cos θ = w2 sin θ cos θ – wx sin θ	(distribute)
y = w sin θ – x tan θ	(divide by w cos θ)

This means that at any vertical line at x, the ladder intersects it at a height of y = w sin θ – x tan θ.  The maximum height of this intersection would be when the derivative ^dy/_dθ = w cos θ – x sec²θ equals zero.  Therefore:

dy/dθ = w cos θ – x sec2θ = 0	(derivative equal to zero)
w cos θ – x/cos2θ = 0	(sec θ = 1/cos θ)
w cos3θ – x = 0	(multiply by cos2θ)
w cos3θ = x	(add x)
cos3θ = x/w	(divide by w)
cos θ = 3√x/w	(cube root)

If cos θ = ³√^x/_w, we can use trigonometric identities to show that sin θ = √(1 – cos²θ) = √(1 – (³√^x/_w)²) and tan θ = sin θ / cos θ = √(1 – (³√^x/_w)²) / ³√^x/_w = ³√^w/_x√(1 – (³√^x/_w)²).  Combining this with our previous equation y = w sin θ – x tan θ, we get:

y = w sin θ – x tan θ	(previous equation)
y = w√(1 – (3√x/w)2) – x 3√w/x√(1 – (3√x/w)2)	(substitute sin θ and tan θ)
y = w√(1 – x2/3w-2/3) – xw1/3x-1/3√(1 – x2/3w-2/3)	(rewrite as exponents)
y = w√(1 – x2/3w-2/3) – x2/3w1/3√(1 – x2/3w-2/3)	(x1x-1/3 = x2/3)
y = w2/3w1/3√(1 – x2/3w-2/3) – x2/3w1/3√(1 – x2/3w-2/3)	(w = w2/3w1/3)
y = w2/3√(w2/3(1 – x2/3w-2/3)) – x2/3√(w2/3(1 – x2/3w-2/3))	(w1/3 = √(w2/3))
y = w2/3√(w2/3 – x2/3) – x2/3√(w2/3 – x2/3)	(distribute, and w2/3w-2/3 = 1)
y = (w2/3 – x2/3)√(w2/3 – x2/3)	(combine like terms)
y = (w2/3 – x2/3)(w2/3 – x2/3)1/2	(rewrite as exponents)
y = (w2/3 – x2/3)3/2	(add exponents)
y2/3 = w2/3 – x2/3	(law of exponents)
x2/3 + y2/3 = w2/3	(add x2/3)

Therefore, the path of the curve that a falling ladder makes is x^2/3 + y^2/3 = w^2/3, where w is the length of the ladder.  In mathematics, this curve is called an “astroid”, which is derived from the Greek word for “star”, because when you graph it in all four quadrants you get a star shape.

Astroid

When you were kid, you may have played with a Spirograph kit, where you can create different designs by rotating geared circles inside another geared circle.

Spirograph Kit

In the same way, an astroid can also be created by following the path of a point on the edge of a circle that is rotated inside another circle that is four times its side.

Creating an Astroid with Circles

To prove this, we can call the radius of the big circle w, the radius of the little circle r, the angle between the x-axis and the segment that contains the two centers of the circles θ, and the angle between segment that contains the two centers of the circles and the segment that contains the center of the little circle to the moving point α. 

Since the big circle is four times the size of the little circle, the center of the little circle is always w – r = 4r – r = 3r away from the origin, so the coordinates of the center of the little circle can be expressed as (3r cos θ, 3r sin θ).  Also since the big circle is four times the size of the little circle, α = 4θ, so the standard angle for (x, y) with respect to the little circle would be 2π – (4θ – θ) = 2π – 3θ.  Putting these two details together, x = 3r cos θ + r cos (2π – 3θ) and y = 3r sin θ + r sin (2π – 3θ).  Simplifying:

x = 3r cos θ + r cos (2π – 3θ)	(x component)
x = 3r cos θ + r cos 3θ	(cos (2π – x) = cos x)
x = 3r cos θ + r (4 cos3θ – 3 cos θ)	(cos 3x = 4 cos3x – 3 cos x)
x = 3r cos θ + 4r cos3θ – 3r cos θ	(distribute)
x = 4r cos3θ	(simplify)
x = w cos3θ	(w = 4r)
x/w = cos3θ	(divide by w)
cos θ = 3√x/w	(cube root both sides)

and:

y = 3r sin θ + r sin (2π – 3θ)	(y component)
y = 3r sin θ – r sin 3θ	(sin (2π – x) = -sin x)
y = 3r sin θ – r (3 sin θ – 4 sin3θ)	(sin 3x = 3 sin x – 4 sin3x)
y = 3r sin θ – 3r sin θ + 4r sin3θ	(distribute)
y = 4r sin3θ	(simplify)
y = w sin3θ	(w = 4r)
y/w = sin3θ	(divide by w)
sin θ = 3√y/w	(cube root both sides)

Finally, substituting cos θ = ³√^x/_w and sin θ = ³√^y/_w into the trigonometric identity sin²θ + cos²θ = 1:

sin2θ + cos2θ = 1	(trigonometric identity)
(3√x/w)2 + (3√y/w)2 = 1	(substitute cos θ and sin θ)
(x/w)2/3 + (y/w)2/3 = 1	(rewrite as exponents)
x2/3 + y2/3 = w2/3	(multiply by w2/3)

which is the equation of an astroid.

There are two remarkable points that should be made about astroids.  First of all, it is amazing that the same curve can be used to describe two completely unrelated situations: a curve made by a falling ladder and a Spirograph design made between two circles.  Secondly, this is yet another geometric formula in the form of xⁿ + yⁿ = zⁿ, where n = ²/₃ for the astroid, n = 2 for Pythagorean’s Theorem, and n = -½ for the radii of three kissing circles and a line (see here).  As mentioned in a previous article, we must admit that so many different formulas of the same form does not describe a chaotic universe of random chance, but rather an orderly universe of intelligent design.