Sacred Geometry Proving pre trig! 8d

[Arugela]

I was doing this for fun along time ago but I can't remember how I found the proof for one thing. Proving that if you have a circle drawn around a circle and keep drawing another one at the intersect you get exactly 6 circle around a base circle without using any modern math. I was doing this to find an alternative basis for the unit circle basically and using only geometry to prove geometry. I found a proof with no modern math but I can't remember how I found it. I may have had to start a secondary pattern on the first and assume a few things until I proved the entire thing. The logic comes out a lot like a certain very old geometry proof book. I think it's greek in origin. It's fun to do and you can definetly proof the beginning of angles using this method for stuff that goes up to proving or developing a value for pie and the basics of triangles in what is basically a form of pre trig.

Has anyone messed with this or know how to prove it. I can't remember and can't refigure it out. Partially because I lack reliable drawing tools to help do it atm and it's hard t improvise those with enough accuraccy to be able to use it as a visual aid beyond the first set or two of circles without. It's a fun way to try to prove geometry related things though. BTW, you cannot use the cirlces position as proof from a drawing. You have to prove it with base logic.

In fact I may have used that old greek book to find the proof... Does anyone know what it is. I forget the name of the book. It uses worded logic to prove a bunch of very basic logical things in mathematics. I think it was written by Euclid.

BTW, sacred geometry is and was a real subject. I think it is known to have been used in ancient religious temples. And if you start playing with it and basically use euclidian logic to prove things. It's potentially very convenient for the development of measurement methods in potentially a very logical or very practical and basic way. The issue is you may need to do some interesting drawings to help develop the logic and it's sometimes hard becuase of paper standards and lack of size. It would basically be a lot more convenient with a lot of space some sand and people moving stuff around a stick potentially. ><

You basically have to prove that 6 circles go around a circle and then use logic to prove other easy things and find proofs working your way up the almost exact stuff Euclid went through, but in different orders(I think I had to combine a few of his proofs to prove it.). It's almost like it's related to an older form of things used as pre trig and possibly related to very old practical building methods. And possibly ways to develop means of early architecture as you could possibly with enough knowledge use it to lay a foundation accurately without a modern laser pointer or other tools. Assuming you don't use something like it with it. If not you could use it to help develop the logic to make the tools.

Either way, it's very coincidental to euclidean logic and the unit circle in all forms. In fact euclidean logic, if you do this sufficiently, is basically giving the answers and a shorthand guide to how to prove these things with sacred geometry/pure geometry proofs.

If no one has ever done this, you start out with basically what turns into a proven equilateral triangle that is then cut in half. You then use basic logic to start defining other base things found in basic geometry and math. I was trying to use it a pure geometric proof up to a certain point to prove things in trig/geometry with basically euclidean logic only. More specifically not using modern info until I developed it myself as a test to try to do it. It makes a very quick potential explanation as to Euclid/Greek obsession with triangles and possible relationship to Egyptian or other older schools of thought or mathematics if that is indeed the case.

I found it. https://mathcs.clarku.edu/~djoyce/java/elements/toc.html

Euclid's elements. 8)

Still, has anyone tried to do this to prove pre trig and work up geometry? It's interesting to attempt. The patterns in sacred geometry basically make a grid to do the things in Euclid's elements. But you have to do a little of the leg work yourself.

I wish I had a hard copy of this book(A good long lasting version at that.). I wonder when this sort of thing will start to disappear. The internet is not a long-standing form of preservation. As much as it would be nice if it was.

Edited July 31, 2018 by Arugela

[Reactordrone]

Here's the Sesame Street version,

 

[Arugela]

I can't see what you posted. It's not showing up for me. Might be a linux thing.

[Reactordrone]

4 hours ago, Arugela said:

I can't see what you posted. It's not showing up for me. Might be a linux thing.

Youtube link, https://www.youtube.com/watch?v=D-op5kqeoEc

 

[Arugela]

Oh, kinky! I don't remember seeing that growing up. 8)

Although I probably did thinking about it.

I don't think I got to proving the rectangles length... I don't remember If I proved the length of the arch from the radius in the flower pattern. If I did I knew the distance. You can also get it from Pythagoras theorem. But I don't know how to develop it off hand from the ground up.

I know you can measure a half radius. I wonder if putting a half radius pattern over the normal one produces a proof. I may have used that at one point. Might help prove the arch vs radius or something similar. Overlaying patterns based on known distances in the pattern might produce easy proofs. it just helps to have really accurate drawing tools and logic to back it up properly.

Edited August 1, 2018 by Arugela

[Zeiss Ikon]

For the  intersecting circles, you can prove you've created a regular hexagon: all the sides are r, and of course that hexagon is made up of six equilateral triangles, also with side length r.

You can see by inspection that the distance from the center of each outer circle to its intersection with the original circle is r, since this is the case for any point on a circle relative to its center.  Using the intersection points as new centers, then, constitutes a geometric construction of equilateral triangles that share the sides forming radii of the original circle, and, from them, a regular hexagon.

Proof by construction, using only geometry, no actual math.

[Arugela]

But you have to prove that the distance between the points is actually the same as r. without logical proof you don't know for certain even if using an extended pattern that seems to show that they are the same. You don't know if they are exact or merely appear to be. If you use several of the euclid's elements proofs you can find a proof they are all exactly 6 and equal in pattern in a circle. Hence all shown disntances in the () pattern are r. But there is an initial problem because you cannot use visuals as proof. It can merely look exact if drawn properly. For all you know it's an infinitesimally small fraction off of a perfect hexagon. As I said, at minimum, there are several proofs from euclids elements that if combined proves logically and absolutely that it is a perfect hexagon. Then also it's known it is a bunch of equilateral triangles. etc, etc.

Just to be clear, you know the initial circle and everything from it's center to it's circumference are r. But you don't know if the outside edges of the hexagon are exactly the same by any visual inspection. Any further drawing of a pattern, which looks like it is a hexagon for many reasons, is not known to be an exact on it's face. Nor can it be proven just by appearance. It requires further logic. Hence the greek love of triangles.

Normally in sacred geometry, you start with a point and draw a circle with the same distance from a point. This is the circumference. This is the same as the definition in Euclid's Elements. After that you can take any point on that circles circumference and draw a new circle with that new circles center point on the circumference of the first circle with a radius of r. then you take an intersect of the two circumferences and draw another circle of the same manner. You repeat this until it goes all the way around. You get a visual, that if drawn well, seem to be exactly 6 circles. But you are left with a logical conundrum of if it is really exactly 6 circles or whether infinitely repeating the pattern PERFECTLY, would eventually create variations. Hence it's not exactly 6 circles but merely appears to be.

You know the distance between the circumference of the first circle drawn on the circumference and the center point of the middle circle do in fact touch because both are drawn with radius r.(excluding visual anomalies from drawing.) All circles drawn on the circumference of circles drawn on the middle circle with radius r touch perfectly the center point of the middle circle. But does the circumference of the sixth circle with the center point on the middle circles circumference perfectly create an intersection with the circumference of the first circle drawn with the center point on the middle circle perfectly on the circumference of the middle circle?(This is assuming you kept drawn circle in the same direction around the middle circle until they collided.)

Edited August 3, 2018 by Arugela

[Zeiss Ikon]

On 8/2/2018 at 8:48 PM, Arugela said:

But you have to prove that the distance between the points is actually the same as r.

You used r in your construction.  No proof needed.  You have r from center to circumference, and two points r apart (straight-line distance, not measured along the curve) on the circumference, therefore (by definition) the three points form an equilateral triangle.  Or are you asking me to prove that six of those make a regular hexagon?  The construction is commonly done with short arcs, but it can be done with full circles; proof of their consistent radius is that they all intersect at the center of the starter circle.

[Arugela]

That is the point. It's a known equilateral triangle because by definition you stuck a same sized circle on the intersect of the other circles. So that 2-3 circles is proof of an equilateral triangle. But how do you know/prove it's a perfect hexagon? You can put the center point on the new intersect on the circumference, but I don't think you have proof of the most important part at that juncture. The very last connecting point of the hexagon. It still needs further proof that it is actually a perfect hexagon. You need knowledge of pie and the calculation of a circumference to proof it's a perfect shape otherwise and that is not usable until developed independently. Else it could be infinitesimally off.

You can go in either direction from the first circle on the middle circles circumference(and infinitely so.). But you don't know for certain that they connect when they visually appear to join at the last circle. You can only prove infinitely going around the circle in either direction independently, without further proof developed. But you never know if they actually connect.

If you get to the sixth circle on the circumference you can choose either of the other circles intersection with the center circle as a starting point to try to prove this. But doing that you do not have proof that the other one intersects perfectly with the other or that the center points are actually the same from the perspective of the outer circles or there intersects between each other. And changing the starting intersection does the same thing in the other direction. Hence the problem. All you can do is say the drawing is close and appears to be linked at that point. But it is not proven. It's literally circular logic.

You do not have to prove a hexagon has six sides. That is part of the definition. What you have to prove is that this is a hexagon to start with. And not what just appears to be dots that get really close to each other and look like a hexagon at the last intersection. It's that last relative intersection between the two outer circles and the inner circle you have to prove are literally on the exact same spot. As I said, current logic only proves the connection as you go around infinitely, but you never know if the things drawn over each other are every connecting from opposite direction. Or even the same direction. This is the natural test here.

Logical test: If you draw three circles in this logic you get 3 equilateral triangles connected in a particular manner that looks like a half hexagon. It basically creates what appears to be a diameter. But how do you prove it's a diameter and what not just appears to be a straight line(AKA, 180degrees which have not yet been established nor that a equilateral triangle is 1/6th of a circle... I don't think.). There could be a simple proof here, but I'm too tired and missing it if so.

Basically, the picture with 3 circles but draw the lines within only one circle.

There might be a simple proof when you draw the second set of circles... I'm not sure if you can proof that larger cross area isn't cutting stuff in half and use that. Or did I miss it already. I'm not sure if the radius going around the outside proves the connection of either last dot. Of course I am really tired so... 8)

Whatever I can't remember about proving it could be really simple. I'll have to read Euclid's elements again. It might jog my mind and help my brain think it out. It always takes me a while to refigure this out. Maybe I'm skipping over the obvious.

Edit: Yea, it could be from things similar to proving that all angles in an equilateral triangle are the same among other things. Euclid dissects this more and several of his proofs go into more detail showing this is the case along with some other logic. But, I think, there are technically a few more bits of logic to fully prove this. I think some of those things are necessary to fully demonstrate the hexagon without any undefined presuppositions. He uses a proof just to establish that the angles of the triangle are the same. Or you can apply several of them to prove this.

 

Edited August 11, 2018 by Arugela

[Zeiss Ikon]

The three angles of any  triangle sum to a flat angle, aka half a full circle -- this is provable, though I don't know that I could reproduce the proof (a simplified, less rigorous form: the three angles must close the triangle, hence all three must combine to return to the original line, hence a flat angle).  In an equilateral, by definition, all the angles are equal, so each is one third of that flat angle; therefore, the three equilaterals constructed as illustrated must produce a flat angle (1/3 flat angle, times three = flat angle).  This proves that the line segment from the leftmost vertex to the rightmost is straight (i.e. no infinitesimal angle at the middle).  And a straight line passing through the center of a circle is the definition of a diameter.  Extend the construction to the full six equilaterals, and you'll have constructed six diameters of the circle, all of which are provable by the three triangles that define them.

Put another way, the construction you have there is halfway through the construction whose goal is to create a regular (inscribed) hexagon.

 

[Arugela]

I found a video form of euclids elements. Much easier to take in. 8) I can't read it anymore.

https://www.youtube.com/watch?v=zdofiH5HncU

 

Edited August 13, 2018 by Arugela