Golden Ratio and Right Angle Triangle

I truly enjoy helping high school and middle school students reach their potentials in solving Mathematical Problems. In this process, I also learn and relearn.

I was working on a right angle triangle problem with some high school students. I realized that there are infinitely many triplets (a,b,c) of side lengths which are in arithmetic progression - for different increments - such as (3,4,5), or any integer multiple of this. We can also get rational increments - again an infinite number of rationals.

On the other hand, for (a,b,c) to be in geometric progression, the ratio can be only one - namely the square-root of (\phi) (or 1.2720196492 - up to 10 decimal places)

Resistances and Golden Ratio

We are given infinite amount of unit resistors (each resistor has a
resistance of 1 Ohm). We can connect these resistors in series or/and in
parallel resulting in a resistor network. For example, you can get a
resistance of 2/3 Ohm by connecting two 1/3 Ohm Resistor networks in series.
To get a 1/3 Ohm Resistor network, we can connect three unit resistors in parallel.
One puzzle is to realize a resistance whose value is the exact golden ratio.

The infinite ladder resistor (with end points A and B) network having a unit resistor
from point A to say Point Y and connect a unit resistor from Y to B. Now repeat the
above construction starting at Point Y. This leads to an infinite ladder network.
and the resistance between points A and B will be the golden ratio.
(as the resistance between A and B is the same as the resistance between Y and B)


To see this, consider the resistor network of unit resistor in serial connection
with a parallel resistor network of 1 Ohm and x Ohms. Let us denote the
resistance between A and B is
x = 1 + x/(1+x)

or x-1 = x/(x+1)

(The author proposed a variation of this problem to IEEE Potential in 1982.
A selected collection of problems including the variation
problem 3-17 appears in
"The Unofficial IEEE Brain-buster Gamebook : Mental Workouts for the Technically
Inclined" by D. R. Mack, Wiley-IEEE Press, 1992.)

With duality (changing resistors in series to parallel and parallel to series) and
scaling, we can realize resistances of all integer powers of the golden ratio as
well as the integer powers of reciprocal of golden ratios!

Redundancy, Robust Design, And Mathematics

Zome
Systems builds upon the magic of the golden ratio.

Its
design reflects redundancy at many levels, a feature that
is

usually common to well-designed systems, and a feature
inherent

to anything mathematical.

As an
example, Zome sticks are both color and shape coded; a

fact
that makes this an excellent tool for teaching math
to

visually impaired students —

see Thinking

Of
Mathematics.

Redundancy can often be a
consequence of the same basic fact

manifesting itself when
viewed from different perspectives. As a

case in point, see
the number of ways that the golden ratio

manifests itself,
both within Zome, and more generally the wider

world we
live in.

tv raman

Some Tidbits

Some Tidbits:

Phi (the golden ratio) plays a very important role in Mathematics,
Sciences, Arts and Zome tool. Please take a look at
http://redsevenone.wordpress.com/
(and click on the link on the right side ZOME - the Answers Before you ask for Zome tool),
and http://en.wikipedia.org/wiki/Golden_ratio (for Phi and Golden Ratio).

If Pi day is celebrated on March 14 (3/14), don't you think Phi day
should be celebrated on 6th of January (1/6)

If Phi is 1.6180339887, then 1/Phi is 0.6180339887 - Do you know why? Because Phi satisfies the algebraic equation Phi^2 = Phi + 1.
If you divide this equation by Phi, you see the answer. This trick was
revealed to me by none other than my co-blogger TV Raman

Math Fun With Zome

Math can be fun, and it can be made even more enjoyable when


elegant mathematical facts are made tangible via concrete models.


Zome Systems --- a polyhedral building kit based on the geometry


of the dodecahedron/icosahedron pair --- can provide many hours


of fun-filled math edutainment.


In this blog, we hope to help to share some of the insights we


have gained over the last few years in the form of easy to


consume byte-sized capsules.


For an in-depth review of many of these facts, see



Visual


Techniques For Computing Polyhedral Volumes
.

Author: T. V. Raman