9 The Mathematics of Spiral Growth. 9.1 Fibonacci’s Rabbits 9.2 Fibonacci Numbers 9.3 The Golden Ratio 9.4 Gnomons 9.5 Spiral Growth in Nature. Gnomons.
9.1 Fibonacci’s Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.5 Spiral Growth in Nature
The most common usage of the word gnomon is to describe the pin of a sundial–the part that casts the shadow that shows the time of day. The original Greekmeaning of the word gnomon is “one who knows,” so it’s not surprising that theword should find its way into the vocabulary of mathematics.
In this section we will discuss a different meaning for the word gnomon.Before we do so, we will take a brief detour to review a fundamental concept ofhigh school geometry–similarity.
We know from geometry that two objects are said to be similar if one is a scaledversion of the other. (When a slide projector takes the image in a slide and blows itup onto a screen, it creates a similar but larger image. When a photocopy machinereduces the image on a sheet of paper, it creates a similar but smaller image.)The following important facts about similarity of basic two-dimensionalfigures will come in handy later in the chapter:
Two triangles are similar if and only if the measures of their respective angles are the same. Alternatively, two triangles are similar if and only iftheir sides are proportional. In other words, if Triangle 1 has sides of length a,b, and c, then Triangle 2 is similar to Triangle 1 if and only if its sides havelength ka, kb, and kc for some positive constant k.
Two squares are always similar.
Two rectangles are similar if their corresponding sides are proportional.
Two circles are always similar.
Any circular disk (a circleplus all its interior) is similar to any other circular disk.
Two circular rings are similar if and only if their inner andouter radii are proportional
In geometry,a gnomonG to a figure A is a connected figure that, when suitably attached to A,produces a new figure similar to A.
By “attached,” we mean that the two figuresare coupled into one figure without any overlap.
Informally, we will describe itthis way: G is a gnomon to A if G & Ais similar to A. Here the symbol“&” should be taken to mean “attached in some suitable way.”
Consider the square S. The L-shaped figure G is a gnomon to the square–when G is attached to S as shown, we getthe square S’.
Consider the circular disk C with radius r. The O-ring G with inner radius r is a gnomon to C. Clearly,G & C form the circular disk. Since all circular disks are similar, C’ is similar to C.
Consider a rectangle R of height h and base b. The L-shaped figure G can clearly be attached to R to form thelarger rectangle. This does not, in and of itself, guaranteethat G is a gnomon to R.
The rectangle R’ [with height (h + x) and base (b + y)]is similar to R if and only if their corresponding sides are proportional, which
This can be simplified to
There is a simple geometric way to determine if the L-shaped G is a gnomonto R–just extend the diagonal of R in G & R. If the extended diagonal passesthrough the outside corner of G, then G is a gnomon; if it doesn’t,then it isn’t.
Let’s start withan isosceles triangle T, with vertices B, C, and D whose angles measure 72º, 72º,and 36º, respectively. On side CD we mark the point Aso that BA is congruent
to BC. (A is the point of intersection ofside CD and the circle of radius BC and center B.)
Since T’ is an isosceles triangle, angle BAC measures 72º and it follows that angle ABC measures 36º. Thisimplies that triangle T’ has equal angles as triangle T and thus
they are similartriangles.
“So what?” you may ask. Where is the gnomon to triangle T? We don’t haveone yet! But we do have a gnomon to triangle T’– it is triangle BAD, labeled G’. After all, G’ & T’
is a triangle similar to T’. Note that gnomon G’ is an isosceles triangle with angles that measure 36º, 36º, and 108º.
We now know how to find a gnomon not only to triangle T’ but also to any72-72-36 triangle, including the original triangle T: Attach a 36-36-108 triangle, G, toone of the longer sides of T.
If we repeat this process indefinitely, we geta spiraling series of ever increasing 72-72-36 triangles.
It’s not toofar-fetched to use a family analogy: Triangles T and G are the “parents,” with Thaving the “dominant genes;” the “offspring” of their union looks just like T (butbigger). The offspring then has offspring of its own (looking exactly like its grand-parent T), and so on ad infinitum.
Example 9.6 is of special interest to us for two reasons. First, this is the firsttime we have an example in which the figure and its gnomon are of the same type(isosceles triangles). Second, the isosceles triangles in this story (72-72-36 and 36-36-108) have a property that makes them unique: In both cases, the ratio oftheir sides (longer side over shorter side) is the golden ratio.These are the onlytwo isosceles triangles with this property, and for this reason they are calledgolden triangles.
Consider a rectangle R with sides of length l(long side) and s (short side), and suppose that the square G withsides of length l is a gnomon to R.
If so, then the rectangle R’ must be similar to R, which implies that their corresponding sides must be proportional (long side of R’ / short side of R´ = long side of R / short side of R):
After some algebraic manipulation the preceding equationcan be rewritten in the form
Since (1) l/s is positive (land s are the lengths of the sides of a rectangle),(2) this last equation essentially says l/s that satisfies the golden property, and (3) the only positive number that satisfies the golden property is , we can conclude that
We can summarize all the above with the following conclusion:
A rectanglewith sides of length l and s (long side and short side, respectively) has a square gnomon if and only if
A rectangle whose sides are in the proportion of the golden ratio is called agolden rectangle. In other words, a golden rectangle is a rectangle with sides l(long side) and s(short side) satisfying l/s = . A close relative to a goldenrectangle is a Fibonacci rectangle–a rectangle whose sides are consecutiveFibonacci numbers.
This rectangle has l = 1and s = 1/.
Since l/s =1/(1/) =,
this is a golden rectangle.
This rectangle has l = + 1and s = .
Here l/s =( + 1)/ .
Since + 1 = 2,this is a golden rectangle.
This rectangle has l = 8and s = 5. This is a Fibonacci rectangle, since 5 and 8
are consecutive Fibonacci numbers. The ratio of the sides isl/s = 8/5= 1.6so this is not a golden rectangle. On the other hand, the ratio1.6 is reasonably close to so we will think of this rectangle as“almost golden.”
This rectangle has l = 89and s = 55 and is a Fibonacci rectangle. The ratio of the sides is l/s = 89/55= 1.61818…,in theorythis is not a golden rectangle. In practice, this
rectangle is as good as golden–the ratio of the sidesis the same as the golden ratio up to three decimal places.
This rectangle is neither a golden nor a Fibonacci rectangle. On the other hand, the ratio of the sides (12/7.44 ≈ 1.613) is very close to the golden ratio.It is safe to say that, sitting on a supermarketshelf, that box of Corn Pops looks temptingly golden.
From a design perspective, golden (and almost golden) rectangles have a special appeal, and they show up in many everyday objects, from posters to cerealboxes. In some sense, golden rectangles strike the perfect middle ground betweenbeing too “skinny” and being too “squarish.”
A prevalent theory, known as thegolden ratio hypothesis, is that human beings have an innate aesthetic bias infavor of golden rectangles, which, so the theory goes, appeal to our natural senseof beauty and proportion.