New Squares On The Block!


Height of an Equilateral Triangle
Height of an Equilateral Triangle
Square with the Area of Three
Square with the Area of Three II
Square with the Area of Twelve
Squares with Areas 1 - 10
The table below shows constructions for squares with areas 1 through 64 using the following four methods and their combinations:

a) Squares with whole number roots (as 1, 4, 9).

b) Squares that are half (or double) the areas of other squares (as 2, 8).

c) Squares whose root is the diagonal of a rectangle (as 5, 10).

d) Squares whose root is the height of a triangle (as 3).

Squares with the area 0f 6 and 7 are combinations of methods b) and d):
6 is half of a square with the area of 12, whose root is the height of a triangle with base = 2, hypotenuse = 4.
7 is (intended to be) quarter of a square with the area of 28, whose root is the height of a triangle with base = 6, hypotenuse = 8

There is an error in this drawing whereby the square that was intended to have an area of 7 instead has an area of 9 3/4. This is because I started with a rectangle of 8x5 rather than 8x6 (8*8 - 6*6 = 28 * 1/2 = 14 * 12 = 7 whereas 8*8 - 5*5 = 39 *1/2 = 19 1/2 * 1/2 = 9 3/4)! The principles are all good though!

Table of Squares

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Geometer's Notebook


Hexagon : Squares
Hexagon with a radius of 1

in relation to squares with areas of 1, 2, 3 & 4.

Triangle unfolding
Notes from "Diagonals of Squares & Rectangles" - work with 5th/6th Grade Students:

Bus notebook:

Student work - much neater than the teacher's!
Copyright: Kayla Nia!


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Secrets of the Pyramids!


Components as Rectangles
Components as Squares
Pyramid Skeleton
Nice Rug?
Pyramid Construction 1
Here's the first idea:

I was looking for a single module that would build both Pyramid and Cube (to allow volume comparisons). Got it down to two so far!

Poor quality photos, sorry!

Inverse Pyramid
Attic Windows
Maltese Cross
Cube Minus Pyramid

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Obsolete Squares Retirement Home



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Golden Sections & Square Roots


Squares 1 through 6
Square 10 - two methods
Square 10 - checking!

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Feeling a bit triumphant about constructing an enneagon from a hexagon, it seemed like an easy next step to construct pentagons from squares!

A pretty fundamental mistake was assuming that equal divisions of a line were also equal divisions of an angle. They are not! Everything (including enneagons) falls apart!

. . . . which leaves me falling back on the lowest common denominator of four and five being twenty, which is not really big news I guess?!?


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Enneagram Riddles & Illusions


Starting to use "cut and fold" to explore ratios - thanks Baravaille!

. . . which leads to a simple extrapolation from a hexagon!

Which is, of course beguiling nonsense!

Simply rotate the yellow triangles 1/3 to the position of the blue triangles and you have your Enneagon!

Base = 3 overlapping Hexagons

Enneagon Beetle, wings closed
Enneagon Beetle, wings closed.

Enneagon Beetle, in flight
Enneagon Beetle, in flight.

All the Angles
Three Overlapping Hexagons with Natural Trisections . . . .

. . . . provide everything needed after all!

There is a slight difference between the edge of an equilateral triangle and the length of the edge of an isosceles triangle with the same height but 1/3 base that I find quite intriguing . . . .

Student work
Student work - all the essential elements!

When AB = x, OC = √[x²-(x/2)²] = √[x²-(x²/4)] = √(3/4x²) = x(√3)/2

When AB = x, OD = √[x²-(x/2)²+(x/6)²] = √[x²-(x²/4)+(x²/36)] = √[(3/4x²)+(1/36x²)] = √[(27/36x²)+(1/36x²)] = √(28/36x²) = √(7/9x²) = x(√7)/3

Now I'm off to attempt pentagons by dividing squares into fifths!





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Projective Geometry


What conditions are required for any line to become either perpendicular or parallel to the horizon?


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Fibonnacci : Harmonic Mean : Golden Section


Fibonacci 13 - 144
Fibonacci 0 - 8
N.B. like the Harmonic Mean, Fibonacci Numbers are a finite sequence.
2 = 1+1
Game over!

Fibonacci - Notes
Numbers converge on the Golden Section (from above and below), doesn't take too long to get to the same decimal places of accuracy as a calculator!

Which repeats what was found with the Harmonic Proportion
- the sequence always terminates at 1:2:

Especially interesting to me is the presence of 8:13 - the very ratio from which, according to Ruland, the Ancient Persian scale was derived!

Next stop: Phyllotaxis!

8 - 13
Warned ya!

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Music II


Completed Projects
Completed Projects II
Thanks Grey!

India Persia Greece & China
The interval given by Ruland for a pentatonic based on the "Ancient Indian Cycle of Sevenths" is not so precise in practice. I was pretty crestfallen when I got to hear my first attempt (based on a 4/7 interval). I knew that I wanted the equivalent of 2.4 modern semitones but I wasn't sure what this would sound like or how to get there in a more precise way. However today I figured that the fifth root of 0.5 (=0.870551) would give me a precise ratio to work with. Actually the 4/7 was not so far off in the second octave, but precision really helps in approaching unusual tones. 

Thing still sounds weird, but I am confident in its weirdness!

Thankfully, after my students worked so hard on all that math, the ratio given for Persia of 8/13 is much more precise. This gives ten equal divisions from which seven are selected to form the scale.

Work in progress . . .
Above shows:

Blue: Mathematical Scale: 1/7 1/5 2/7 1/3 3/7 1/2 4/7 2/3 5/7 4/5 6/7
which is a variation of the GiQuan based around the interval of 1/7 and leaving enough room between frets for football team stickers! 

Green: Persian Scale = Equal intervals based on Cycle of Sixths = 8/13

The above would be a nice project - four fretboards based on different mathematical intervals (with possibly and "even-tempered" one alongside for comparison).

But they also show the following as easily achievable using "natural harmonics":


Do = open

Re = 8/9

Mi = 4/5

Fa = 3/4

So = 2/3

La = 3/5

Ta = 5/9

Do = 1/2

Mixolidyan but with two octaves gives all modes as previously:

. . . . also "chromatic", with only the flat ninth and major seventh elusive

(what do you mean "Quite right too!"?)

Do = open

Ra = ?

RE = 8/9

Ma = 5/6

Mi = 4/5

Fa = 3/4

Fi/Sa = 7/10

So = 2/3

Le = 5/8

La = 3/5

Ta = 5/9

Ti = ?

Do = 1/2

Here's a chromatic fretboard without those pesky semitones "Ra" (flat nine) and "Ti" (Major seventh) - all you need are two strings tuned a fifth apart.

This is really quite lovely - a fretboard with all the chromatic options (bar changing key!) and no fractional intervals in the double digits . . . . if it works!


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Previously published:

All 28 blog entries

Here are some of the ideas that I have been exploring.

Please let me know if there's anything you want to correct/find interesting/would like a copy of - or to purchase/participate in etc.

 Philip James GuestLos Angeles, CA310.383.2327