Million Ways to Cut a Square in 4

MILLION WAYS TO CUT A SQUARE IN 4.

A 4 STEP PROCEDURE OF APPLIED

CREATIVITY. David Solomon Hadi - Chief Strategist

Financial Services

Rock Star Consulting Group

www.rockstarconsultinggroup.com

http://www.rockstarconsultinggroup.com/

INTRODUCTION

This is a demonstration of a four step procedure that generates large number of creative possibilities of achieving a clearly defined logical end.

The method is for applying creative thinking skills to projects while remaining with-in logical bounds of project.

However, it can be used for artistic purposes too.

The example given here is visual and simple for ease of understanding.

Million Ways to Cut a Square in 4. A 4 Step procedure of Applied Creativity. David Solomon Hadi - Chief Strategist Financial

Services Rock Star Consulting Group (www.rockstarconsultinggroup.com)






4 STEPS OF THE METHOD

1. Ask what is needed.

Make it refined. Keep only and only what is needed.

2. Ask if means to reach the ends are restricted or not?

3. Generate possible means to achieve the end keeping in mind 1 and 2.

4. Ask if in step 3 there is a repeating pattern, something frequent.

If yes, go back to step 3 with additional restriction of not repeating it.








MOTIVATION OF STEP 1 AND 2

1. Ask what is needed. Make it refined. Keep only and only what is needed.

In projects sometime instructions may not be well defined. This may hurdle some possible ways to achieve the end.

Also by refining one becomes clear in one’s own thinking and over come hurdles in generating patterns.

Same holds for question “2. if means to reach the end are restricted or not?”








MOTIVATION OF STEP 4

4. Ask if in step 3 there is a repeating pattern,

something frequent. If yes, go back to step 3 with

additional restriction of not repeating it.

While generating possibilities, easiest way is to

find newness is to find repetition and remove it.

This, as would be clear in example, leads to easy

and yet powerful pattern generations.








EXAMPLE: CUT SQUARE IN FOUR. STEP 1

Task:

Cut a square in four pieces.

1. What is exactly is needed? What is “pieces”? Is

it defined by area or circumference? Should

pieces be equal?

Cut a square in four equal areas and not

circumference.








EXAMPLE: CUT SQUARE IN FOUR. STEP 2

Is there any restriction as how to cut and how not

to cut?

No.

Is it possible to use any form of lines or any mean

to cut?

Yes








EXAMPLE: CUT SQUARE IN FOUR. START

OF STEP 3 AND 4 LOOP.

The Area of each of these is easy to calculate.

Area of rectangles = length * width.

It should be simply ¼ of area of square.

Area of square = X * X [ X = one side of square]








EXAMPLE: CUT SQUARE IN FOUR.

Is there a repeating Pattern?

Yes, it is all straight lines.

Change this.









Area of these are relatively difficult to calculate.

But with due care cuts can be made so area of ¼

is achieved.

If lines cut square in symmetric parts, then

intuitively area would be same.









One can generate many patterns with curves as

in A (left).

There can be many shorter sine waves.

After exhausting several curves, one can replace

repeating curves with Zig Zags (B right).

There can several forms of Zig Zags.

Exhaust all possible forms of Zig Zag pattern.









Now one may ask what was common is all of preceding designs?

Single shape divides square in four.

Every time a straight line, a curve or a zig zag repeats itself.

Also lines always touch ends e.g. corners or opposite edges.

How to avoid it?









Area of circle is Pi * r * r [radius = r]

Area of square = X*X [X = side of square]

Solve for r for the left diagram as:

¼ * X*X = ¼ * Pi*r*r

Solve for r for the rest of diagram as:

¼ * X*X = ½ * Pi*r*r









Area of tri-angle is ½ base * height (for diagram

in left)

Square is first divided into two halves(two

triangles for left and rectangles for rest).

Removing ¼ of square from half of square would

leave a ¼ of square area for each piece.









Now lines do not repeat. Circles add creative

touch. Also edges problem is partially solved.

One may develop many such figures.

What would be repeating?

Every time a circle and line would repeat

themselves.









For far left

Pi*radius*radius (circle) = ½ base * height (Tri Angle) = height * width (rectangle) = ¼ * X*X (Square) [X is one side of square]

Similar calculations may be done for far right.

Cut out the shapes in square and remaining area would be ¼ of square.









Now two repeating shapes are replaced by three.

This is maximum because four cuts are needed.

Several such shapes can be drawn.

What would be repeating? Million Ways to Cut a Square in 4. A 4 Step procedure of Applied Creativity. David Solomon Hadi - Chief Strategist Financial







WHAT IS REPEATING ?

All these shapes are drawn with compass or

ruler.

These shapes are nice and regular.

Their areas are easy to calculate.

Regular, easy to calculate, nice, drawn with

ruler.

This is repeating . How to break this?









Break all repetition with:

Irregularity

Area of an irregular object is difficult to

measure but not impossible.

See http://www.basic-mathematics.com/area-of-

irregular-shapes.html



http://www.basic-mathematics.com/area-of-irregular-shapes.html
















Area of each of these can be ¼ and remaining

part of square would have ¼ area of original

square.

How many different irregular shapes can one

generate? Hundreds? Thousand?

Millions.








FURTHER WORK

We are working on this project and would like to receive your feed back.

Our aim is “development of simple and clear step wise procedures to assist applied creative thinking.”

Kindly share your ideas. Contact person responsible for this project: [email protected]

You are always most welcome.



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Million Ways to Cut a Square in 4