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Synergetics In The Plane

By

Ian Beardsley

Copyright © 2020 by Ian Beardsley


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Table of Contents

Developing An Intuitive Understanding of Basic Mathematics………..5

Synergetics In The Plane…………………………………………………..11


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We begin with the the basic concepts at the at the foundation of mathematics presented in
such a way that they open up like a flower, each concept a flowering of that which precedes it,
then go into what I call Synergetics in the Plane.

In doing my research in establishing the relationships between biological life, artificial

intelligence (AI) and the planets, I have noticed recurrent mathematical themes. I just noticed
that if they are all taken together they can comprise a mathematical system that can be called
Synergetics In The Plane.

Synergetics by Buckminster Fuller has always been remarkable to me since I first learned of it
when I was in High School, I see it as the creation of a mathematical system along with a
language that make geometric analysis more accessible to a broader range of people while
creating a language for the general populace that brings us more up to par with being a
technologically advanced civilization.

Synergetics makes math easier by eliminating the need for fractions by making it so that
computations can be done with whole numbers.

This work Synergetics in the Plane is relationships in two dimensions, all of which are used in
doing math in three dimensions but in contrast to Fuller’s Synergetics does not make use of
whole numbers, but rather is centered most often around irrational numbers such as square
root 2, square root 3, square root 5,….where irrational numbers are unending decimals that
cannot be written as the ratios between two whole numbers. After twenty years of working on
the aforementioned project, I have been able to write this out off the top of my head.

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Developing An Intuitive Understanding of Basic Mathematics

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Formulas Derived from the Parallelogram

Remarks. Squares and rectangles are parallelograms that have four sides the same
length, or two sides the same length. We can determine area by measuring it either in
unit triangles or unit squares. Both are fine because they both are equal sided, equal
angled geometries that tessellate. With unit triangles, the areas of the regular polygons
that tessellate have whole number areas. Unit squares are usually chosen to measure
area.

Having chosen the unit square with which to measure area, we notice that the area of a
rectangle is base times height because the rows determine the amount of columns and
the columns determine the amount of rows. Thus for a rectangle we have:

A = bh
Drawing in the diagonal of a rectangle we create two right triangles, that by symmetry
are congruent. Each right triangle therefore occupies half the area, and from the above
formula we conclude that the area of a right triangle is one half base times height:

1
A= bh
2
By drawing in the altitude of a triangle, we make two right triangles and applying the
above formula we find that it holds for all triangles in general.

We draw a regular hexagon, or any regular polygon, and draw in all of its radii, thus
breaking it up into congruent triangles. We draw in the apothem of each triangle, and
using our formula for the area of triangles we find that its area is one half apothem times
perimeter, where the perimeter is the sum of its sides:

1
A= ap
2
A circle is a regular polygon with an infinite amount of infitesimal sides. If the sides of a
regular polygon are increased indefinitely, the apothem becomes the radius of a circle,
and the perimeter becomes the circumference of a circle. Replace a with r, the radius,
and p with c, the circumference, and we have the formula for the area of a circle:

1
A= rc
2
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We define the ratio of the circumference of a circle to its diameter as pi. That is pi=c/D.
Since the diameter is twice the radius, pi=c/2r. Therefore c=2(pi)r and the equation for
the area of a circle becomes:

A = πr 2
(More derived from the parallelogram)

Divide rectangles into four quadrants, and show that

A. (x + a)(x + b) = x 2 + (a + b)x + ab

B. (x + a)(x + a) = x 2 + 2a x + a 2
A. Gives us a way to factor quadratic expressions.

B. Gives us a way to solve quadratic equations. (Notice that the last term is the square
of one half the middle coefficient.)

Remember that a square is a special case of a rectangle.

There are four interesting squares to complete.

1) The area of a rectangle is 100. The length is equal to 5 more than the width multiplied
by 3. Calculate the width and the length.

2) Solve the general expression for a quadratic equation, a x 2 + bx +c =0

3) Find the golden ratio, a/b, such that a/b=b/c and a=b+c.

1 2
4) The position of a particle is given by x = vt + at . Find t.
2
Show that for a right triangle a 2 = b 2 + c 2 where a is the hypotenuse, b and c are legs.
It can be done by inscribing a square in a square such that four right triangles are made.

Use the Pythagorean theorem to show that the equation of a circle centered at the
origin is given by r 2 = x 2 + y 2 where r is the radius of the circle and x and y the
orthogonal coordinates.

Derive the equation of a straight line: y=mx+b by defining the slope of the line as the
change in vertical distance per change in horizontal distance.
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Triangles

All polygons can be broken up into triangles. Because of that we can use triangles to
determine the area of any polygon.

Theorems Branch 1

1. If in a triangle a line is drawn parallel to the base, then the lines on both sides of the
line are proportional.

2. From (1) we can prove that: If two triangles are mutually equiangular, they are similar.

3. From (2) we can prove that: If in a right triangle a perpendicular is drawn from the
base to the right angle, then the two triangles on either side of the perpendicular, are
similar to one another and to the whole.

4. From (3) we can prove the Pythagorean theorem.

Theorems Branch 2

1. Draw two intersecting lines and show that opposite angles are equal.

2. Draw two parallel lines with one intersecting both. Use the fact that opposite angles
are equal to show that alternate interior angles are equal.

3. Inscribe a triangle in two parallel lines such that its base is part of one of the lines and
the apex meets with the other. Use the fact that alternate interior angles are equal to
show that the sum of the angles in a triangle are two right angles, or 180 degrees.

Theorems Branch 3

1. Any triangle can be solved given two sides and the included angle.

c 2 = a 2 + b 2 − 2abcos(C )
2. Given two angles and a side of a triangle, the other two sides can be found.

a b c
= =
sin(A) sin(B) sin(C )
3.Given two sides and the included angle of a triangle you can find its area, K.

K=(1/2)bc(sin(A))
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4.Given three sides of a triangle, the area can be found by using the formulas in (1) and
(3).

Question: what do parallelograms and triangles have in common?

Answer: They can both be used to add vectors.

Trigonometry

When a line bisects another so as to form two equal angles on either side, the angles
are called right angles. It is customary to divide a circle into 360 equal units called
degrees, so that a right angle, one fourth of the way around a circle, is 90 degrees.
The angle in radians is the intercepted arc of the circle, divided by its radius, from which
we see that in the unit circle 360 degrees is 2(pi)radians, and we can relate degrees to
radians as follows: Degrees/180 degrees=Radians/pi radians

An angle is merely the measure of separation between two lines that meet at a point.

The trigonometric functions are defined as follows:

cos x=side adjacent/hypotenuse

sin x=side opposite/hypotenuse

tan x=side opposite/side adjacent

csc x=1/sin x

sec x=1/cos x

cot x=1/tan x

We consider the square and the triangle, and find with them we can determine the
trigonometric function of some important angles.

1 2
Square (draw in the diagonal): cos(45)∘ = =
2 2

3 1
Equilateral triangle (draw in the altitude): cos(30∘) = ; cos(60∘) =
2 2
Using the above formula for converting degrees to radians and vice versa:
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π π
30∘ = radians; 60∘ = radians
6 3
The regular hexagon and pi

Tessellating equilateral triangles we find we can make a regular hexagon, which also
tessellates. Making a regular hexagon like this we find two sides of an equilateral
triangle make radii of the regular hexagon, and the remaining side of the equilateral
triangle makes a side of the regular hexagon. All of the sides of an equilateral triangle
being the same, we can conclude that the regular hexagon has its sides equal in length
to its radii.

If we inscribe a regular hexagon in a circle, we notice its perimeter is nearly the same as
that of the circle, and its radius is the same as that of the circle. If we consider a unit
regular hexagon, that is, one with side lengths of one, then its perimeter is six, and its
radius is one. Its diameter is therefore two, and six divided by two is three. This is close
to the value of pi, clearly, by looking at a regular hexagon inscribed in a circle.

The sum of the angles in a polygon

Draw a polygon. It need not be regular and can have any number of sides. Draw in the
radii. The sum of the angles at the center is four right angles, or 360 degrees. The sum
of the angles of all the triangles formed by the sides of the polygon and the radii taken
together are the number of sides, n, of the polygon times two right angles, or 180
degrees. The sum of the angles of the polygon are that of the triangles minus the angles
at its center, or A, the sum of the angles of the polygon equals n(180 degrees)-360
degrees, or

A=180 degrees(n-2)

With a rectangular coordinate system you need only two numbers to specify a point, but
with a triangular coordinate system --- three axes separated by 120 degrees -- you need
three. However, a triangular coordinates system makes use of only 3 directions,
whereas a rectangular one makes use of 4.

A rectangular coordinate system is optimal in that it can specify a point in the plane with
the fewest numbers, and a triangular coordinate system is optimal in that it can specify
a point in the plane with the fewest directions for its axes. The rectangular coordinate
system is determined by a square and the triangular coordinate system by an
equilateral triangle.
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Synergetics In The Plane

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The foundation of our synergetics in the plane will begin with the regular tessellators, polygons
that are equal sided and equal angled throughout, where poly means many and gon means
side. Regular means equal sides and equal angles throughout. There are three regular polygons
that tessellate, which means they can cover a surface without leaving gaps. They are the 3-
gon, 4-gon, and 6-gon, which can be called the trigon, tri meaning three, the tetragon, tetra
meaning four, and the hexagon, hex meaning six, which are the equilateral triangle, square, and
regular hexagon respectively, where the square is a special case of a rectangle. They tessellate
as shown in fig 1.

In Buckminster Fuller’s Synergetics, the tetrahedron is the fundamental unit of volume,

tetrahedron meaning four-faced and it looks as in fig 2.

And, he chose this as the fundamental unit because if he defined it as having a volume of one,
then the rest of the ensuing solids would have volumes of a whole number of tetrahedrons,
thus eliminating the need for fractions. However, in our 2-dimensional synergetics if we define
the area of an equilateral triangle as 1, the regular hexagon does have a whole number area of
six, but the square does not as shown in fig 3.

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The dual of an equilateral triangle is a regular hexagon where the dual is obtained by
connecting dots in the center of each triangle as in fig 4.

Thus in 2-dimensional synergetics, or synergetics in the plane, we do not deal with whole
numbers. Rather we often deal with irrational numbers like those involving 2 , 3 , 5 and
fractions such as 3/2 or 1/2. At the foundation of these numbers are the special triangles as
shown in fig 5 and fig 6.

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If we want to look at the regular pentagon (penta meaning 5) we find it does not tile the surface
alone: But requires the equilateral triangle (colored in) in fig 7, a so called Archimedean
tessellator.

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However, the regular pentagon does tile a surface in three dimensions, however because it
encloses a volume it is finite as demonstrated in fig 8 a dodecagon.

However, the regular tessellator the square, which tiles the plane infinitely, tiles in three
dimensions as well, finitely, as in the cube in fig 9.

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However, the equilateral triangle tiles the 3D finitely as well, for example in the tetrahedron fig
10.

But, the regular pentagon is very important because it is the source of the golden ratio phi (Φ)
or its conjugate lower case phi (ϕ) where Φ = ϕ + 1, and Φ = 1/ϕ.

Detour: The angles in an n sided regular polygon are

A + B + C = 180∘

◯= 360 degrees

∢ = 180∘(n − 2)

As explained in fig 11.

θ = 180(n − 2)/n

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The angles in a regular pentagon are 108 degrees.

Returning from detour: The golden ratio is in the regular pentagon as outlined in fig 12.

We can derive the golden ratio as such (refer to fig 13):

a b
= = Φ

b c
a = b + c

ac = b 2

c = a − b

a(a − b) = b 2

a 2 − a b − b 2 = 0

a2 a
− − 1 = 0

b2 b
a2 a 1 1
− + = 1 +

b2 b 4 4
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(b)
2
a a 1 1
− + =1+

b 4 4

(b)
2
a a 1 4 1 5
− + = + =

b 4 4 4 4

(b 2)
2
a 1 5
− =

a 1 5
− =

b 2 2
5+1
Φ=

b 5−1
ϕ= =

a 2

Thus we establish the following important equation for synergetics in the plane (fig 14 and fig
15)…

(6)
π
2cos(30∘) = 2cos = 3

(4)
π
2cos(45∘) = 2cos = 2

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(5)
π 5+1
2cos(36∘) = 2cos = = 1.618

The equation being:

(n)
π
Nn = 2cos

n = (4,5,6)

N4 = 2

5+1
N5 =

2
N6 = 3

Detour: Interestingly if we compare the same number of molecules of air to the same number
of molecules of water we have the golden ratio.

Air is about 75% nitrogen gas (N2) and 25% oxygen gas (O2).

N=14.01 g/mol, and O=16.00 g/mol. Thus,…

N2 = 2(14.01) = 28.02

O2 = 2(16.00) = 32.00

0.75(28.02)=21 and 0.25(32.00)=8 giving air=29.0 g/mol

Thus we have:

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H2O = 2(1.01) + 16.00 = 18.02g/m ol

air 5+1
= 1.61 ≈ Φ = = 1.618

H2O 2
Return from detour: Much in the same way that resolution of definition of an image on a
computer screen is characterized by the number of pixels per inch, I think maybe the resolving
power of space is in the number of points per unit length that may exist and, that, this in turn is
the factor that determines the clarity of Reality. We might guess that it is on the an order
determined by Planck’s constant (ℏ). Thus, it may look like (fig 16):

But, since in this work we are working in two dimensions, or the plane, we have (fig 17):

And, we see that if the points are homogeneously distributed (distributed evenly in both row
and column) they are the tessellations of squares which are regular polygons:

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If we want to tessellate equilateral triangles in the plane resolved into pixels like we did with the
squares, we find we have to stagger the dots by 1/2 from row to row where as with the squares
in our earlier example they were homogeneously distributed. Thus, for the tessellation of
triangles that are regular, we have (fig 19):

Which becomes…

As Fuller said, only systems of triangles are inherently stable. This because each side is
stabilized by two opposing sides.

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This allows for a rectangular coordinate system through which we can specify any point in the
plane with two numbers (fig 21):

However, Fuller says in Synergetics that Nature employs trigonal coordination, which looks like
fig 22):



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In our synergetics in the plane we will say that 180∘ or π radians are both open, and closed
stable (fig 23):

We see that 180∘ is two right angles (fig 24) and the sum of the angles in a triangle are 180∘ .
That is a + b + c = 180∘ by way of opposite angles are equal.

We will say the net force in a geometry is given by the sum of the cross-product that between
the vectors that comprise it. That is:

i⃗ j⃗ k ⃗
Ax Ay Az = A Bsin(θ )

Bx By Bz

The following two sketches do precisely this…


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The Author

This book copyright © 2020 by Ian Beardsley