Ellipse and Ring

The third little geometrical gem is about a ring and an ellipse. An ellipse is the shape that a plate or anything circular appears when viewed at an angle.

Here are two circles, one inside the other. The yellow and red areas define an outer ring. Also the orange and red parts form an ellipse (an oval).

If the ring is very narrow, the two circles are similar in size and the ellipse has a much bigger area than the ring.

If the inner circle is very small, the ellipse will be very narrow and the outer ring will be much bigger in area than the ellipse.

So the question is

The answer is again when the inner radius is 0.618 of the larger one, the golden ratio.

This is quite easy to prove using these two formulae:

The are of a circle of radius r is π r²
The area of an ellipse with "radii" a and b (as shown above) is πab

(Note how that, when a = b in the ellipse, it becomes a circle and the two formulae are the same.)
So the outer circle has radius a, the inner circle radius b and the area of the ring between them is therefore:

Area of ring = π (b² – a²)

This is equal to the area of the ellipse when

π (b² – a²) = π a b
b² - a² - a b = 0

If we let the ratio of the two circles radii = b/a, be K, say, then dividing the equation by a² we have

K² - K - 1=0

which means K is either Phi or –phi. The positive value for K means that b = Phi a or a = phi b.

The equation of an ellipse is

(x/b)² + (y/a)² = 1

When a = b, we have the equation of a circle of radius a(=b)

:

(x/a)² + (y/a)² = 1 or
x² + y² = a² as it is more usually written.

You might have spotted that this equation is merely Pythagoras' Theorem that all the points (x,y) on the circle are the same distance from the origin, that distance being a.

Note 79.13 A Note on the Golden Ratio, A D Rawlings, Mathematical Gazette vol 79 (1995) page 104.
The Changing Shape of Geometry C Pritchard (2003) Cambridge University Press paperback and hardback, is a collection of popular, interesting and enjoyable articles selected from the Mathematical Gazette . It will be of particular interest to teachers and students in school or indeed anyone interested in Geometry. The three gems above are given in more detail in the section on The Golden Ratio.

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The Fibonacci Spiral and the Golden Spiral

The Fibonacci Squares Spiral

On the Fibonacci Numbers and Golden Section in Nature page, we looked at a spiral formed from squares whose sides had Fibonacci numbers as their lengths.

This section answers the question:

The Golden section squares are shown in red here, the axes in blue and all the points of the squares lie on the green lines, which pass through the origin (0,0).
Also, the blue (axes) lines and the green lines are each separated from the next by 45° exactly.
The large rectangle ABDF is the exactly the same shape as CDFH, but is (exactly) phi times as large. Also it has been rotated by a quarter turn. The same applies to CDFH and HJEF and to all the golden rectangles in the diagram. So to transform OE (on the x axis) to OC (on the y axis), and indeed any point on the spiral to another point on the spiral, we expand lengths by phi times for every rotation of 90°: that is, we change (r,theta) to (r Phi,theta + π/2) (where, as usual, we express angles in radian measure, not degrees).

So if we say E is at (1,0), then C is at (Phi,π/2), A is at (Phi², π), and so on.
Similarly, G is at (phi,–π/2), and I is at (phi², –π) and so on because phi = 1/Phi.

The points on the spiral are therefore summarised by:

If we eliminate the n in the two equations, we get a single equation that all the points on the spiral satisfy:

or

For ordinary (cartesian) coordinates, the x values are y values are generated from the polar coordinates as follows:

x = r cos(theta)
y = r sin(theta)

which we can then use in a Spreadsheet to generate a chart as shown here.
Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals. They also have the property that a line from the origin to any point on the curve always finds (the tangent to) the curve meeting it at the same angle.
Another name is a logarithmic spiral because the angle of any point from the x axis through the origin is proportional to the logarithm of the point's distance from the origin.

To see that the Fibonacci Spiral here is only an approximation to the (true) Golden Spiral above note that: at its start there are two squares making the first rectangle but the true golden spiral above has no "start" those two squares make a rectangle 1x2 but all rectangles in the true spiral are true Golden Rectangles 1xPhi. All the other rectangles on the right are ratios of two neighbouring Fibonacci numbers and are therefore only approximations to Golden Rectangles

The Golden or Phi Spiral

In the spiral above, based on Fibonacci squares spiralling out from an initial two 1x1 squares, we noted that one quarter turn produces an expansion by Phi in the distance of a point on the curve from the "origin". So in one full turn we have an expansion of Phi⁴.
In sea-shells, we notice an expansion of Phi in one turn, so that not only has the shell grown to Phi times as far from its origin (now buried deep inside the shell). Also, because of the properties of the golden section, we can see that the distances measured on the outside of the shell also have increased by Phi and it is often easier to measure this distance on the outside of a shell, as we see in the picture here on the left.
In this case, the equation of the curve is

Notice that an increase in the angle theta of 2 π radians (360° or one full turn) makes r increase by a factor of Phi because the power of Phi has increased by 1.
I will call this spiral, that increases by Phi per turn, the Golden Spiral or the Phi Spiral because of this property and also because it is the one we find in nature (shells, etc.).

Click on the Spreadsheet image to open an Excel Spreadsheet to generate the Golden Spiral in a new window.

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Trigonometry and Phi

What is trigonometry?
We can answer this by looking at the origin of the word trigonometry.
Words ending with -metry are to do with measuring (from the Greek word metron meaning "measurement"). (What do you think that thermometry measures? What about geometry? Can you think of any more words ending with -metry?)
Also, the -gon part comes from the Greek gonia) meaning angle. It is derived from the Greek word for "knee" which is gony.

The prefix tri- is to do with three as in tricycle (a three-wheeled cycle), trio (three people), trident (a three-pronged fork).
Similarly, quad means 4, pent 5 and hex six as in the following:

• a (five-sided and) five-angled shape is a penta-gon meaning literally five-angles and
• a six angled one is called a hexa-gon then we could call
• a four-angled shape a quadragon
  (but we don't - using the word quadrilateral instead which means "four-sided") and
• a three-angled shape would be a tria-gon
  (but we call it a triangle instead)
  "Trigon" was indeed the old English word for a triangle.

So trigonon means "three-angled" or, as we would now say in English, "tri-angular" and hence we have tri-gonia-metria meaning "the measurement of triangles".
With thanks to proteus of softnet for this information.

Phi and Trig graphs

Here are the graphs of three familiar trigonometric functions: sin x, cos x and tan x in the region of x from 0 to π/2 (radians) = 90°:
The graphs meet at

• the origin, where tan x = sin x
• in the middle, where sin x = cos x i.e. where tan x = 1 or x = 45° = π/4 radians
• at another angle where tan x = cos x

What angle is at the third meeting point?

tan x	= cos x and, since tan x = sin x / cos x , we have:
sin x	= (cos x)2
	=1-(sin x)2 because (sin x)2+(cos x)2=1.
or	(sin x)2 + sin x = 1

and solving this as a quadratic in sin x, we find
sin x = (–1 + √5)/2 or
sin x = (–1 – √5)/2
The second value is negative and our graph picture is for positive x, so we have our answer:

On our graph, we can say that the intersection of the green and blue graphs (cos(x) and tan(x)) is where the red graph (sin(x)) has the value phi [i.e. at the x value of the point where the line y = phi meets the sin(x) curve].

Is there any significance in the value of tan(x) where tan(x)=cos(x)?

Yes. It is √phi = √0·618033988... = 0·786151377757.. .

Here's how we can prove this.

Take a general right-angled triangle and label one side t and another side 1 so that one angle (call it A) has a tangent of t. By Pythagoras's Theorem, the hypotenuse is √(1+t2). So we have: For all right-angled triangles: tan A = t cos A = 1 √(1 + t2) sin A = t √(1 + t2)

We now want to find the particular angle A which has tan(A) = cos(A).
From the formulae above we have:

Squaring both sides:

Multiply both sides by 1 + t²

:

If we let T stand for t² then we have a quadratic equation in T which we can easily solve:

SinceT is t² it must be positive, so the value of T we want in this case is

Since T is t² then t is √phi.
Since 1 + Phi = Phi² then the hypotenuse √(1+T²) = Phi as shown in the triangle here.
From the 1, √Phi, Phi triangle, we see that

 

In the triangle here, it is angle A:
A = 38.1727076° = 30° 10' 21.74745" = 0.1060352989.. of a whole turn = 0.666239432.. radians.
The other angle, B has its tangent equal to its cosecant (the reciprocal of the sine):
B = 90° – A = 51.8272923..° = 51° 49' 38.2525417.." = 0.143964701.. of a whole turn = 0.90455689.. radians.

Notice how, when we apply Pythagoras' Theorem to the triangle shown here with sides 1, Phi and root Phi, we have

which is one of our classic definitions of (the positive number) Phi.....
and we have already met this triangle earlier on this page!

The next section looks at the other trigonometrical relationships in a triangle and shows that, where they are equal, each involves the numbers Phi and phi.

Notation for inverse functions

A common mathematical notation for the-angle-whose-sin-is 0.5 is arcsin(0.5) although you will sometimes see this written as asin(0.5).
We can prefix arc (or a) to any trigonometrical function (cos, cot, tan, etc.) to make it into its "inverse" function which, given the trig's value, returns the angle itself.
Each of these inverse functions is applied to a number and returns an angle.
e.g. if sin(90°)=0 then 0 = arcsin(90°).

What is arccos(0.5)?
The angle whose cosine is 0.5 is 60°.
But cos(120°)=0.5 as does cos(240°) and cos(300°) and we can add 360° to any of these angles to find some more values!
The answer is arccos(0.5) = 60° or 120° or 240° or 300° or ...
With all the inverse trig. functions you must carefully select the answer or answers that are appropriate to the problem you are solving.

"The angle whose tangent is the same as its cosine" can be written mathematically in several ways:

Can you see that it can also be written as arccos( tan(A) )= A?

More trig ratios and Phi (sec, csc, cot)

In a right-angled triangle if we focus on one angle (A), we can call the two sides round the right-angle the Opposite and the Adjacent sides and the longest side is the Hypotenuse, or Adj, Opp and Hyp for short.

You might wonder why we give a name to the ratio Adj/Opp (the tangent) of angle A but not to Opp/Adj. The same applies to the other two pairs of sides: we call Opp/Hyp the sine of A but what about Hyp/Opp? Similarly Adj/Hyp is cosine of A but what about Hyp/Adj?

In fact they do have names:

• the inverse ratio to the tangent is the cotangent or cot i.e. Adj/Opp; cot(x)=1/tan(x)
• the inverse ratio to the cosine is the secant or sec i.e. Hyp/Adj; sec(x)=1/cos(x)
• the inverse ratio to the sine is the cosecant or cosec or sometimes csc i.e. Hyp/Opp; csc(x) = 1/sin(x)

You'll notice that these six names divide into two groups:

• secant, sine, tangent
• cosecant, cosine, cotangent

and show another way of choosing one representative for each of the 3 pairs of ratios (x/y and y/x where x and y are one of the three sides).

Here is a graph of the six functions where the angle is measured in radians:

This extended set of graphs reveals two more intersections involving Phi, phi and their square roots: e.g.

• if A is the angle where cos(A) = tan(A) then sin(A) = phi and cosec(A) = Phi;
  The value of A is A = 38.172..° = 0.666239... radians;
• if B is the angle where sin(B) = cot(B) then cos(B) = phi and sec(B) = Phi;
  The value of B is 51.827..° = 0.904556... radians.

Notice that A and B sum to 90° - as we would expect of any two angles where the sine of one is the cosine of the other.

Some Results in Trigonometry L Raphael, Fibonacci Quarterly vol 8 (1970), pages 371-392.

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Some Results in Trigonometry, Brother L Raphael, The Fibonacci Quarterly vol 8 (1970) pages 371 and 392.