Mathematics Dude

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The 345 Code

The sacred 345 triangle has many mathematical and geometric lessons to teach.

The 345 code is fascinating and shows how whole numbers are connected by an elaborate set of geometry. Follow my current progress at folding circles.

Below we can see how the numbers 1-9 are connected.

Then we can see how the 345 triangle is connected to the golden ratio and the double Spiral of Theodorus. All important triangles are given a unique colour.

Next we can see how the 345 triangle is connected to the square, equilateral triangle and the infinite spiral. Below we can see the whole numbers 1-10.

Next we see the importance of √5 and how it connects to the square and octagon.

Below the Metallic Means 1-10 and the amazing connection to the golden ratio.

This is just the tip of an infinite geometric iceberg.

The 345 triangle was revered by the Egyptians, each side given a divine name.

Also we see the geometry of the 345 triangle in many Egyptian Hieroglyphs.

The middle pyramid (Khafre) at the Giza plateau encodes a 345 triangle.

Basic Numbers using cosmic geometry.

Below the Metallic Means 1-14 showing cosmic (±) geometry and 345 triangle.

This hidden pattern in the Metallic Ratios links the 345 with the Spiral of Theodorus.

The amazing 345 triangle.

The geometry of the whole.

Folding Circles. The amazing phi.

More phi (golden ratio) using cosmic geometry.

The flower of life using cosmic geometry.

The Kingdom is within.

The amazing 345 triangle is the foundation stone. (Right angle)

Cosmic (±) geometry and shapes.

Remember, remember. (Covenant of Salt, the Sacred Hoop). Weighing scales.

The Amazing geometry of the Squared Circle.

A picture speaks a thousand words, this next picture 1001.

The geometry of the sacred 345 triangle and the number 7 (3+4=7)

This new geometry method actually makes it easier to understand.

Absolutely amazing geometry is a real brain buster.

Metallic V Organic means coming out from the magical Pythagorean spiral.

Jain pi or phi-pi is most fascinating, no other angle has this symmetry.

More geometry regarding the sacred 345 triangle.

Be water, my friend….

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The Amazing Golden Ratio (Phi)

The Golden ratio is maybe the greatest number, lets explore why?

The golden ratio can be found in both the Hexagon and the Pentagon.

Phi (golden section) triangle and the 345 triangle have a unique relationship.

And in order to understand phi we must look in detail at this relationship. Below showing the most important angles related to the 345 triangle.

This relationship becomes very sophisticated very quickly.

Below we can see the complicated world of phi, showing relationships to many interesting numbers and angles.

Yes it is complicated.

Both the first (GOLD) and fourth (NICKEL) metallic mean is related to Phi.

The metallic means using cosmic geometry. Again phi triangle is mean 1 and 4.

Here we can see that there is also a relationship between the golden ratio and the number sequences (Fibonacci and Lucas).

Number Sequences (Fibonacci, Lucas and Pell)

We can also relate phi to Binet’s formulas, as shown below.

It is because of its unique fractal nature. (incommensurability, phase conjugation, harmonic convergence) Only Phi has this fractal nature.

Below we can see the golden Spiral, along with the golden rectangle. Light and Magnets both share this fractal nature.

Here we can see the symmetry of the magnetic flux lines. (ferro-cell image)

Light and Magnetism both use the magical ratio.

The colours of the visible light spectrum and Phi. This electro-magnetic universe has a favourite number and it appears to be a golden.

A geometrical figure, where each part has the same character as the whole.

They are seen in things like snowflakes in which patterns recur at different scales.

If you know of any interesting phi relationships, please comment.

Art by Ken Wheeler, Dan Winter, Clay Taylor and me.

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Sacred Geometry 101

It is assumed that (x) is always ≥ 1 and the (1/x) is always ≤ 1.

The ‘cosmic’ equation

Linking the ‘cosmic’ equation with Pythagoras.

Linking the ‘cosmic’ equation with trigonometry and the trigonometric functions.

Simplifying the tan relationship.

more stuff

more stuff

The Triangles

The relationship between the red and blue triangles.

More stuff.

Example A

Visualising the triangles using the ‘cosmic’ equation.

Substituting x₁ (from above) into the green triangle.

Scale by x₁ and substitute cos. Using circles to perform square-roots (more information).

Example B

More stuff.

And then..

Folding Circles

These pages are old and confusing. 🐇 (I am sorry for any inconvenience.)

An aerospace engineer messing around with numbers and geometry on a journey through space/time with only my compass to guide me. Lets start folding circles.

There are 2 different kinds of folding fish (square fish) and (cosmic fish). The square fish are shown above and fold into 1/8th of a square based pyramid, with examples from the square root spiral shown below.

And below the trusty 345 pyramid. Infinite pyramids in the spiral of Thedorus.

The cosmic fish are detailed below, these are a lot more sophisticated.

There are an infinite amount of these folding fish contained within the double spiral.

Exploring the amazing geometry encoded into of the pyramids. Giza analysis.

This is the double spiral which has a fascinating relationship with the 345 triangle.

The double Pythagorean spiral is one of the most fascinating things i found on my journey, it has two triangles for every triangle in the well known square-root spiral.

On my journey I found the amazing connection between the sacred 345 triangle and golden ratio as shown below, using the method I named cosmic (±) geometry.

Cosmic (±) geometry is sophisticated but basically every angle can be represented by a number (y) and each right angled triangle has two angles and two (y) values.

This is a powerful method as shown with the Metallic means and Organic means.

I didn’t realise how much Cosmic (±) Algebra there would be, interestingly the etymology of the word is from Arabic al-jabr ‘the reunion of broken parts’.

Most of this fascinating geometry comes from following the golden ratio (phi).

As well as folding circles I have also been collecting interesting fish for folding.

Below, more folding fish and the sacred geometry of the whole numbers (2-10).

I use colours to utilise my whole brain and simplify this sophisticated method.

I feel i am getting closer to reaching the foundations of this method and the folding fish geometry makes it simpler and easier to explain. (need geometry machine V2)

The 345 triangle is the foundation stone of geometry used to create a right angle.

The sacred cross, the geometry of the square (45 degrees) and the cross.

Cosmic geometry and folding circles, squares, triangles into folding fish.

It amazes me that all this geometry came from the humble sacred 345 triangle.

I have been studying this method and folding circles for many years. I started off my journey following the golden ratio (phi) through the geometric universe and now find myself studying the whole numbers, trying to reach the foundations of the method.

Every abstract whole number actually has a complex geometry associated with it and all whole numbers are connected together as shown in the square-root spiral.

My main interests include mathematics (specifically geometry), symbolism and art.

“In the beginning God created the heaven and the earth. And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.And God said, Let there be light.” Genesis 1

The Geometry of Atlantis

Let no one ignorant of geometry enter here. The concentric circles of Atlantis.

Reminds me of the geometry of the Pythagorean double square root spiral, that connects the whole numbers, the fragment below contains the numbers. (1-10)

It contains 2 right angled triangles for every triangle in the spiral of Theodorus.

The geometry of Atlantis is amazing and the fishing (infinite fish) is great.

Each concentric circle has its own specific geometry and exquisite architecture.

If you enjoy fishing then Atlantis is the place for you, endless fish in pristine still waters.

Every fish you catch can be folded into a solid as shown above with the 345 triangle.

The colouring of each fish is beautiful, particularly our prize winning goldfish.

Our famous Atlantis pink salmon is served on plates designed by the legendary Plato.

Metallic V Organic Means

We can use cosmic (±) geometry to create a right angled triangle using every Metallic Mean, as shown in the examples below where y is equal to each mean/ratio.

Below we can see the Metallic means (1-14) and some other interesting triangles.

These three interesting triangles are shown below in blue(c), red and white(m).

And below we can see how the triangles fit together below, the blue (345 triangle) and the red triangle are linked together, as well as the blue and yellow triangles.

The purple triangles are both linked to the white (m) lunation triangle.

The amazing 345 triangle seems to show up all over the place hence the 345 code.

As shown above, the angles are very close to the mentioned geometry which is ideal for comparison (wink, wink, nudge, nudge). Finally the idea of organic means/ratios.

From this point on, things get a little bit confusing. Seriously, don’t break your brain.

Its colour coded, so you can use the right side of your brain as well as the left side. When you are a master you will be able to use the bit in the middle also. The lunation triangle seems to symbolise a link the moon (13) and sun (12).

Most of this geometry seems to have its root in the Pythagorean spiral above, as it contains an infinite amount of triangles with each pair being linked together.

We can see above how each triangle interacts and links with the next.

Some triangles contain more gold than others, from the Lucas number sequence.

Some triangles contain other rare minerals like silver-ratio and the bronze-ratio.

The Geometry Machine

I have made many geometry machines and geometry animations made with JavaScript over the years. The last version is ancient and needs updating.

These are all available for you to play with at codePen (Folding Circles)

The geometry machine (v3).

I would like to create a new version in python as JavaScript is not good enough. If there is anybody out there in the cosmos that knows python and has a keen interest in geometry and the golden ratio and would like to collaborate on the next version, leave a comment or send a smoke signal, carrier pidgeon, bat man sign, smoke me a kipper.

The geometry machine uses a method I call cosmic (±) geometry and allows us to get to the roots of geometry and the triangle, trigonometry and the Pythagoras theorem.

I can then use the code to create moving geometry animations.

more gif animations (GifyU, ).

all based on the one method.

and then..

The Covenant of Salt

Following the golden thread through the mathematical universe using (±) method.

A deep dive into the complex geometry of the golden ratio.

This very deep dive into a very sophisticated method.

Is colour coded to help understand.

And will take many years to mentally digest.

Please do not break your brain, it will take many years to understand.

It took me seven years to write it down.

I am still trying to understand it and I am being guided by the (w)holy spirit.

The amazing 345 triangle is the foundation stone of this method.

This method shows how whole numbers are connected together.

Cosmic Geometry

Using what I call cosmic mathematics, we can (for instance) detail the geometry of a triangle without using trigonometry or the Pythagoras theorem. And that is good.

Based on a simple equation, it adds a level of sophistication to the normal method, giving us a deeper level of understanding. Taking into account the usually ignored negative principle is a bit of a mind-bender, so, patience is a virtue.

We can link this equation with Pythagoras’s theorem.

Adding some more information. Below the 345 triangle and more cosmic geometry.

and more geometry.

More phi.

achieve: The golden ratio is one of the most interesting cosmic number.

Below we can see how we can use cosmic geometry to generate the Metallic means.

Interestingly we can generate the Pythagorean triples, without using the Pythagoras theorem. Below we can see that using cosmic geometry provides us with a lot more useful information when compared with other methods.

Pythagorean Triples using Cosmic Geometry

As we can see the 345 Triangle is very special case of Pythagorean triple.

The 345 triangle is interesting for many reasons, below we can see its amazing symmetry (platonic solids) and its relationship with the golden ratio.