## Thursday, January 31, 2008

### Golden Ratio and Right Angle Triangle

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)

## Monday, January 14, 2008

### Resistances and Golden Ratio

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!

## Saturday, May 19, 2007

### 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.

## Friday, May 4, 2007

### 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