This web page is reproduced from the author’s book Principles of nature; towards a new visual language, WA Roberts P/L Canberra. 2003. [Used with permission of the author] edited and reformatted for the web, with some additional commentary for the sake of contextual clarity.


There exists a further double-resonance of the 120° angle in both the Cosine Rule (in which 2cosC= -1) and also in the scale structure of the three regular polygons that can tile the plane without gaps (one of these being the hexagon whose internal angles are 120°). These synchronicities suggest a third special class of triangle in which one internal angle equals 120°, and for which class we are justified in expecting a similar associated Pythagoras-like theorem (in light of scale-structure theory as discussed thus far).

Introducing another new class of triangle (as fundamental as the right triangle) and its associated theorem*

Let us thus define a triangle in which one angle equals 120-degrees to be a co-eutrigon (the prefix “co” signifying complement of the eutrigon).

Definition of a co-eutrigon

Let triangle ABC be any co-eutrigon. We define angle C to be the 120° angle, a and b to be the ‘legs’ or sides adjacent to angle C , and c to be the hypotenuse or side opposite angle C.

co-eutrigon_labelled

Let a co-eutrigon be a triangle with one 120° internal angle.

The Co-eutrigon theorem

Introduction

This, and the Eutrigon Theorem (geometric form..), may be new. The closest theorem I have found to the geometric forms of the Eutrigon Theorem … and the Co-eutrigon Theorem … is the Napoleon Theorem…. Via a scale-structure approach including a consideration of the geometry in terms of relative units of area, I was able to deduce algebraic forms of theEutrigon and Co-eutrigon Theorems. The resultant equations ( ab = a2 + b2c2 for eutrigons, and ab = c2a2b2 for co-eutrigons) were confirmed by, and geometrically reinterpreted, the Cosine Rule (in terms of relative units) when angle C = 60° and 120° respectively.

cosine_rule_scalene_triangle

The Co-eutrigon theorem:
geometric form

The area of any co-eutrigon [shaded in the diagram below] is equal to the area of the equilateral triangle on the hypotenuse c minus the sum of the areas of the equilateral triangles on legs a and b.

co-eutrigon_and_surrounding_equilateral_triangles_labelled

above, Figure C-ET1

A ‘look-and-see’ proof:

co-eutrigon_theorem_visual_proof_and_scale_structure

Figure C-ET2

Figure C-ET2 shows a resonant scale structure
which visually proves the Co-eutrigon Theorem.

Detailing the Co-eutrigon theorem proof

co-eutrigon_theorem_diamond_scale_structure_labelledFrom Figure 83 it can be seen that :

1) Triangles marked “Q” are congruent co-eutrigons since each contains one 120° angle, and their respective sides are equal.

2) Triangles A, B. and C are equilateral.

3) A and B are the equilateral triangles on the legs of co-eutrigon Q, and C is the equilateral triangle on its hypotenuse.

4) The overall diamond shape is a rhombus consisting of an upper and lower equilateral triangle of identical area.

5) Point “4” means that, (expressed in terms of areas):

4A + 4Q + B = 3A + 3Q + C

Thus A + Q + B = C,

Or, Q = C – A – B

The Co-eutrigon Theorem is proved.

Figure C-ET3         

Can Q be any co-eutrigon?

For reasons similar to those given in the case of the Eutrigon theorem, Q,…can represent any co-eutrigon, proving the theorem true for all co-eutrigons. We may summarise this reasoning as follows:

The shape of any co-eutrigon is specified by the ratio of its leg lengths, a/b. If we let the shorter of a eutrigon’s legs be a, then the ratio a/b is always in the range 0 < a/b < (or equal to) 1. It can be seen from an examination of [the] figure … that every ratio of a/b is possible in the diagram: a can be vanishing small or can be any value up to and including a = b. Thus every possible shape of co-eutrigon can be accommodated in the diagram without altering the geometric relations, and thus the theorem holds true for all co-eutrigons.

Co-eutrigon Theorem: algebraic form

The geometric form of the Co-eutrigon theorem states: the area of any co-eutrigon (i.e. a triangle in which one angle is 120°) is equal to the area of the equilateral triangle on its hypotenuse ‘c’ minus the combined areas of the equilateral triangles on legs ‘a’ and ‘b’ [see figure C-ET1 above]. How can this be expressed algebraically, that is, as an equation?

Earlier we determined the area of an equilateral triangle in etu (which is simply p2 where p is the side length) but we have not yet determined the equation for the area of a co-eutrigon in etu. However, there is a beautiful synchronicity with the equation for the eutrigon’s area which follows from a correspondence of the triangle altitudes between eutrigons and co-eutrigons. This again reflects the complementary relationship between the two classes of triangle.

It is well-known and easily proven that triangles of the same base and same altitude have the same area. This means that there is a surprising resonance between a eutrigon’s and co-eutrigon’s areas —if legs a and b (i.e. the sides adjacent to the defining angle) are equal then it follows that they have the same area, namely their product, ab (as expressed in etu),[see figure below].

co-eutrigon_and_eutrigon_with_same_altitudes

These two triangles have the same area because their bases and altitudes are the same.

We may demonstrate the complementarity of these triangles more clearly by placing them as [below],

eutrigon_and_co-eutrigon_placed_side_by_side

Same triangles as above but here placed side-by-side to highlight their complementarity. The area of each triangle is identical when the bases b are equal and share the same 60° altitude a.

So the area of a co-eutrigon expressed in etu is the same as for the corresponding eutrigon of the same respective leg lengths and thus simply the product of the co-eutrigon’s leg lengths, ab.

We can therefore state the algebraic form of the Co-eutrigon Theorem in terms of the new relative units of area (etu) as,

 The area of a co-eutrigon (given in etu) = ab = c2a2b2

This follows from [Figure C-ET3] and the ‘area equation’, Q = C – A – B.

As with the Eutrigon theorem’s algebraic form discussed earlier, the Co-eutrigon Theorem is also consistent with the Cosine Rule and is the same as that rule for the special case when angle C = 120°. … Since 2CosC = -1, the Cosine Rule reduces to c2 = a2 + b2 + ab and, rearranging terms, we obtain the Co-eutrigon theorem form above,

ab = c2a2b2

Including the Pythagorean equation, we now have three Pythagoras-like equations and their corresponding geometric theorems (through the induction of relative units). Given that aand b are the sides adjacent to the defining angle (e.g. the 90° angle of a right triangle), and c is the hypotenuse or side opposite the defining angle, the three equations are:

  • a2 + b2 = c 2 . . . . Pythagoras’ Theorem for case when C = 90°
  • ab = a2 + b2c2 . . . . . . Eutrigon Theorem; C = 60°
  • ab = c2a2b2 . . . . . . Co-eutrigon theorem; angle C = 120 E

These three equations and their associated geometric forms exactly correspond (via their respective stipulated internal angle) to the three regular polygons (equilateral triangle, square, and hexagon) which can uniformly tile the flat plane without gaps.

This then completes a scale structure of not only three algebraic theorems but of their corresponding resonant geometric theorems, and it is reasonable to conjecture, I feel, that when recognised and implemented as a complete scale structure within mathematical practice, and utilised in ‘resonant application’, that significant advances may follow in number theory and in our understanding of the foundations needed for a new visual language and music.

For example, the whole sub-discipline of trigonometry may now be re-examined in light of the new geometric understanding of the above equations (which includes the critical notion ofrelative units).

Number theory too is likely to be extended via the key of the relative unit, and will call into question the very foundations of number and the meaning of ‘integers’—of how they are written or represented, and of new operations, properties, and transformations that may now be discovered and made possible.

This web page is reproduced from the author’s book Principles of nature; towards a new visual language, WA Roberts P/L Canberra. 2003. [Used with permission of the author] Edited and reformatted for the web, including animated diagrams and additional commentary by the author.


Introduction

Scale structure theory has implications not only to the units and measurement of areas but also to triangle classification as we have seen. Particularly pertinent to the theorem presented here on this page is the implementation of the equitriangular unit of area (etu) as we defined earlier. We will also later use our new knowledge of the area of a eutrigon (in terms of these relative units of area, etu) in the algebraic interpretation of the geometric construction (figure ET2) below — a form of resonant scale structure (in the terminology of the new theory) — that visually proves the theorem.

the Eutrigon Theorem

—Geometric Form

The area of any eutrigon (shaded in figure ET1) is equal to the sum of the areas of the equilateral triangles on its legs a and b, minus the area of the equilateral triangle on its hypotenuse, c.

eutrigon_with_surrounding_equilateral_triangles

Geometric form of Eutrigon Theorem

The area of any eutrigon is equal to the sum of the areas of the equilateral triangles on its legs a and b, minus the area of the equilateral triangle on its hypotenuse, c.

(W. Roberts, 2003, p.122)

A look-and-see proof

Note: all triangles which appear equilateral are equilateral.

eutrigon_theorem_visual_proof_and_scale_structure

Detailing the proof (see figure ET3)

eutrigon_theorem_visual_proof_and_scale_structure_labelledFrom Figure ET3 it can be seen that:

1) Triangles marked “Q” are congruent eutrigons since each contains one 60° angle, and their respective sides are equal.

2) Triangles A, B, and C are equilateral

3) A and B are the equilateral triangles on the legs of eutrigon Q, and C is the equilateral triangle on its hypotenuse. >

4) The overall diamond shape is a rhombus consisting of an upper and lower equilateral triangle of identical area.

5) Point “4” means that, expressed in terms of areas,
A + B + 2Q = C + 3Q,
and thus, Q = A + B – C

therefore the Eutrigon Theorem is proved.

 

Figure ET3

Now, some readers may not be satisfied that in the ‘look & see’ proof given above, that eutrigon Q may be any eutrigon . Before we prove this more rigorously let us imagine whether it is so.

Imagine an animation of equilateral-triangle C rotating while maintaining its equilateral shape but varying in size such that its vertices ‘slide along’ the outer equilateral triangle it is touching…[I’ve provided such an animated version below and added some colour to jazz it up a bit! It auto-loops a few times with a pause in between. If it has already been through its routine by the time you get to this part of the page, just click the ‘refresh button’ in your browser window, and it will start over.]

eutrigon_theorem_animation

Figure ET4
(animated, © 2003 copyright W Roberts. All rights reserved)

For an elaboration of the proof that “Q” in Figure ET3 covers every possible eutrigon shape.

See here for PDF [173kb] of geometric form of Eutrigon Theorem from the book (W. Roberts, Principles of Nature: towards a new visual language, 2003, pp. 122-125)

This web page is quoted from the author’s book Principles of nature; towards a new visual language, WA Roberts P/L Canberra. 2003. [Used with permission of the author] – minor editing for the web


In the regular division of the plane, there exists a scale structure of only three distinct regular polygons which can tile the plane to infinity in every direction without gaps. They are the square, the equilateral triangle and the hexagon.

three_regular_polygons

These gapless tiling patterns (known as tessellations of the plane) are demonstrated in the next figure (Figure 6.7)

regular_division_of_plane

Figure 6.7

There are only three regular polygons whose sides and angles are equal and which can tile the plane without gaps: the square, the equilateral triangle, and the hexagon

That only these three regular polygons can tile the plane without gaps and in self-similar manner is a fact that has been known by geometers for centuries and is easily proven (Dunham, 1994, pp. 108-111). What is perhaps not recognized or undervalued is the potential significance of such a fact. [This appears to be symptomatic of a problem in current cultural attitudes to knowledge and research, and will be briefly addressed later.]

We should therefore expect that these three shapes contain within themselves and within this triadic scale, important clues to the structure and form of the infinity which they divide. Just as the time it takes for an echo to return reflects the distance of a rock face on the other side of a gorge, so these shapes ought to reflect something about the nature of the infinity they divide. And because they divide this infinity in a self-similar way, that is, because they divide this infinity into integral units, we expect these shapes will reflect something important about the whole-numbers themselves. More particularly, they should reflect something important about special sorts of whole number, because, of all the regular polygons (i.e. having equal side-lengths and internal angles), there are just three which can tile the plane without gaps. Since polygons are shapes which occupy area, and since areas can be expressed numerically in a multiplicative manner (length x breadth), we should expect numbers of a symmetrical composition in the form p x p (that is, p2) may be connected to these particular regular polygons in certain special ways. That is exactly what we find. These special sorts of whole number we currently know as the squares.

Since polygons are shapes which occupy area, and since areas can be expressed numerically in a multiplicative manner (length x breadth), we should expect numbers of a symmetrical composition in the form p x p (that is, p2) may be connected to these particular regular polygons in certain special ways. That is exactly what we find. These special sorts of whole number we currently know as the squares.

The line of reasoning may be schematically summarized as in figure 6.7 below.

sequence_of_steps_for_square_numer_evolution

Figure 6.7

Line of reasoning leading to the scale structure of the plane

The square property of numbers of the form p2 has been widely known for more than two thousand years. But on the preceding page (click button b ) we have seen a less familiar triangular representation of the ‘square number’ 4. We need to show if this triangular characteristic is a property of just some of the square numbers, or all of them.

Do all square numbers have a corresponding triangular form?

It has long been known that the squares can be generated by the successive addition of the members of the set of odd numbers (1+3=4, 1+3+5=9, 1+3+5+7=16, …). Geometrically, these odd numbers may represent integral areas which are added to a given shape, and such added shapes were called gnomons by the Greeks if they could regenerate the original figure (see figure 6.8). Clearly, it can be seen that for each unitary increase in the side-length of either squares or equilateral triangles, the gnomon consists of successive odd numbers of component self-similar shapes. Each successive gnomon contains two more units than its predecessor. And therefore each successive gnomon includes successive members of the set of odd numbers and can therefore form successive numbers of the form p2. Thus every square number can just as easily be represented in equilateral triangular form.

gnomons

Figure 6.8 Illustrating gnomons which when added to a given shape reproduce the original

What about hexagons?

A significant oversight and omission is thus revealed at the fundamental level of the notion of the unit in mathematics—at the very least, units pertaining to the measurement of areas (also refer: Schrödinger’s observation)— including the concept of units as taught in primary, secondary, and even (sometimes) tertiary education

[In light of the equally-triangular nature of the so-called ‘square numbers’] ..is it not curious then, that in the history of mathematics, squares have served the role of ‘general-purpose units-of-area’, it would appear by default rather than design*? We have applied square plugs to nearly every shaped hole under the mathematical sun. The notion of units-relative-to-wholes has not been taken up seriously perhaps because issues of comparison have tended to focus on differences more than similarities or ‘resonances’. In the comparative measurement of areas, we have inducted the square as the default unit of area and a universal arbiter of comparison (‘this area is so-many-square units whereas that area is a lesser number of equivalent square units, and is therefore smaller’). It is ironic because comparisons involve relating something to something else, and thus a choice of units relative to the system concerned must ultimately be simpler and more powerful.

We have applied ‘square plugs’ to nearly every shaped hole under the mathematical sun.

As will be shown in the following pages, the application of relative units to mathematical and spatial problems allows a deeper understanding of qualitative differences as well as a simpler and more resonant connection of geometric problems to number theory and analysis.