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Truth is Triangular

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The triangle is a great representation of dialectic or synthesis. The circle is a great representation of logic or number.

Pure logic is pure number and pure number is only one number, despite appearances. All number is a modification of 1. Don't be deceived by our ten-digit positional system. It's convenient, as we have ten digits. But ten digits is one quantity. (More on that later...)

Logos, however, is not circular but triangular (or spiral, but that's another thread). For logos, or word, is not only the unification of qualia, but also of other words. Logos is an transcendentally synthetic, but only because it relates us to the incidental, by unifying qualia.

Philosophy is nothing but this synthesis, by which the transcendental is revealed/abstracted, including the synthetic process itself, which is both transcendental and incidental. (symbolized quite well by Christ...)

The cross (+) expresses the same synthesis, but that's another thread....

@Reconstructo,

Reconstructo;133937 wrote:

The triangle is a great representation of dialectic or synthesis. The circle is a great representation of logic or number.

If it's an equilateral triangle, then it can be described as a circle with three being the value of pi.

@ughaibu,

ughaibu;133944 wrote:

If it's an equilateral triangle, then it can be described as a circle with three being the value of pi.

Thanks! I didn't know that. That's actually perfect, because logos is as digital as it is continuous, and you just helped me tie the two together more....(but are you rounding pi down? or something else?....

@Reconstructo,

Reconstructo;133946 wrote:

are you rounding pi down? or something else?....

No, the value is equal to the number of sides. This is the case for any convex polygon, they dont need to be equilateral, but that reduces the eccenticity of the idea. The interesting point is that pi for circles is between three and four, not infinitely large, as one would expect if a circle is a polygon with an infinite number of sides.

@ughaibu,

ughaibu;133948 wrote:

No, the value is equal to the number of sides. This is the case for any convex polygon, they dont need to be equilateral, but that reduces the eccenticity of the idea. The interesting point is that pi for circles is between three and four, not infinitely large, as one would expect if a circle is a polygon with an infinite number of sides.

Would you mind providing the formula? Are we talking about area, sum of the sides? I'm quite interested in this. Any info would be appreciated...

@Reconstructo,

Reconstructo;133952 wrote:

Would you mind providing the formula? Are we talking about area, sum of the sides? I'm quite interested in this. Any info would be appreciated...

Think of a king on the central point of a chess board. He can get to anywhere on the edge in four moves, so for him the board is a circle with radius four. As there are four sides, the value of pi is four. Using the normal formulae for circles he calculates the area of the board as sixty four and the circumference as thirty two, which is correct. Any convex polygon can be divided into a system of quadrilaterals by joining the centres of the edges to the centre of the polygon, then the normal formulae for cicles will give the area in quadrilaterals and the circumference in sides of quadrilaterals.

@ughaibu,

ughaibu;133953 wrote:

As there are four sides, the value of pi is four.

But isn't pi an irrational constant?

@Reconstructo,

Reconstructo;133955 wrote:

But isn't pi an irrational constant?

Pi is the ratio of the circumference to the diameter, in the case of circles it's a transcendental number, in the case of any other convex polygon it's equal to the number of sides, if the ratio is considered as above. Different values can be generated according to how the ratio is defined.

@ughaibu,

ughaibu;133960 wrote:

Pi is the ratio of the circumference to the diameter, in the case of circles it's a transcendental number, in the case of any other convex polygon it's equal to the number of sides, if the ratio is considered as above. Different values can be generated according to how the ratio is defined.

Are you sure it's pi, and not just pi as a metaphor? I appreciate your help, so don't take this quote as unfriendliness....the issue is just quite important to me...

Quote:

π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation (see the table for its representation in some other bases). The constant is also known as Archimedes Constant, although this name is rather uncommon in modern, western, English-speaking contexts. Many formulae from mathematics, science, and engineering involve π, which is one of the most important mathematical and physical constants.[5]

@Reconstructo,

Reconstructo;133966 wrote:

Are you sure it's pi, and not just pi as a metaphor? I appreciate your help, so don't take this quote as unfriendliness....the issue is just quite important to me...

If pi is the ratio of the circumference to the diameter, then there is nothing in this definition that restricts pi to circles. Of course that restriction generally applies, but that's because describing other convex polygons in terms of this ratio is unusual behaviour. If you dont want to use the letter "pi", feel free to use something else, but the idea is easier to grasp if the terminology is consistent.

@ughaibu,

ughaibu;133973 wrote:

If pi is the ratio of the circumference to the diameter, then there is nothing in this definition that restricts pi to circles. Of course that restriction generally applies, but that's because describing other convex polygons in terms of this ratio is unusual behaviour. If you dont want to use the letter "pi", feel free to use something else, but the idea is easier to grasp if the terminology is consistent.

I feel you! I'm not a stickler. You were using pi metaphorically, which is fine. I just needed to know, as pi is quite important to me. The thread Digital Time in Analog Space is all about why pi must be irrational. Shall we use some other variable? How about z?

@ughaibu,

ughaibu;133973 wrote:

If pi is the ratio of the circumference to the diameter, [...]

It's not clear to me how one ought to define 'diameter' for a figure other than a circle. Also, the symbol pi in this context already has a fixed meaning, so it would be better to use some other name, to avoid confusion. Rho, say.

On the other hand, a regular polygon, just like a circle, has a circumference, L, and an area, A. (The most general case, including both regular polygons and circles as special cases, would probably be a "rectifiable Jordan curve", but don't quote me on that!)

How should we define 'rho', so that it comes out as pi in the case of a circle? For a circle of radius R:

A = pi * R^2

L = 2 * pi * R

therefore:

pi = (L^2) / (4A)

so we could define, for any rectifiable Jordan curve (or whatever!), of length L, enclosing an area A:

rho = (L^2)/(4A)

For a regular polygon with n sides, we have:

L = 2n * sin(pi/n)

A = n * sin(pi/n) * cos(pi/n)

therefore:

rho = n * tan(pi/n)

For n = 3 (equilateral triangle):

rho = 3 * sqrt{3} = 5.20 approx.

For n = 4 (square):

rho = 4

For n = 6 (regular hexagon):

rho = 2 * sqrt{3} = 3.46 approx.

For any rectifiable Jordan curve whatsoever:

rho >= pi

but this is hard to prove - it is the famous "Isoperimetric Inequality". I hope I've stated it correctly. It seems so (and Wikipedia even uses the same notation, although it doesn't use the term "rectifiable Jordan curve"):

Isoperimetric inequality - Wikipedia, the free encyclopedia

@Twirlip,

Twirlip;134008 wrote:

For any rectifiable Jordan curve whatsoever:

rho >= pi

but this is hard to prove - it is the famous "Isoperimetric Inequality".

Okay, but you're talking about a different matter than I was.
There's a nice proof of the isoperimetric theorem in this book: Amazon.com: Maxima and Minima Without Calculus (Dolciani Mathematical Expositions) (9780883853061): Ivan Niven, Lester H. Lance: Books

@ughaibu,

ughaibu;134035 wrote:

Okay, but you're talking about a different matter than I was.
There's a nice proof of the isoperimetric theorem in this book: Amazon.com: Maxima and Minima Without Calculus (Dolciani Mathematical Expositions) (9780883853061): Ivan Niven, Lester H. Lance: Books

That does indeed look like a nice book - I'll add it to my list of books to try to get hold of when I've recovered more of my interest in mathematics (although that's taking so long that it looks as if I'll have to hope for reincarnation!).

And I agree that the Isoperimetric Inequality is off-topic.

But what, then, were you saying about circles and polygons and their "diameters" which was on-topic? You still haven't explained what you mean by "diameter", and it was this gap that I was trying to fill with something that I could at least understand.

(Also, BTW, I can't find your introductory post to the forum.)

@Twirlip,

Twirlip;134135 wrote:

But what, then, were you saying about circles and polygons and their "diameters" which was on-topic?

Whether it was on or off topic isn't clear, to me, because I dont understand what this thread is about. However, the thread begins by distinguishing a representation of "dialectic and synthesis" from one of "logic and number", I wanted to point out that these representations can be merged, so dont really distinguish anything.

Twirlip;134135 wrote:

You still haven't explained what you mean by "diameter", and it was this gap that I was trying to fill with something that I could at least understand.

Pi is most easily related to the circle as the ratio of diameter to circumference, but it would be more general to state this as the ratio of twice the radius to the circumference, because in the case of polygons the ratio is similarly expressed in terms of the circumference and apothem. A polygon with an even number of sides has a diameter, in the sense that the circle has, but a polygon with an uneven number of sides doesn't, so I would define the diameter as twice the apothem.

Twirlip;134135 wrote:

(Also, BTW, I can't find your introductory post to the forum.)

I haven't yet posted one.

@ughaibu,

ughaibu;134261 wrote:

Whether it was on or off topic isn't clear, to me, because I dont understand what this thread is about. However, the thread begins by distinguishing a representation of "dialectic and synthesis" from one of "logic and number", I wanted to point out that these representations can be merged, so dont really distinguish anything.

I completely disagree. The distinction is essential. If you don't see why, give it some thought. Words aren't (only) numbers. Mathema is tautological. Logos is synthetic.

This is why truth is triangular, as the triangle is a great visual metaphor for synthesis.

Number can teach us nothing, for number is only one number. It doesn't matter if you call this number "one". Number is essentially pure equation.

Without contraries there can be know time or progress. If words were like numbers, culture could not evolve. Synthesis is the creation of new meaning. Equation is not.

@ughaibu,

ughaibu;134261 wrote:

A polygon with an even number of sides has a diameter, in the sense that the circle has, but a polygon with an uneven number of sides doesn't, so I would define the diameter as twice the apothem.

I wasn't familiar with the term 'apothem', but I see that it means the same as 'inradius', i.e. the radius of the inscribed circle. That clarifies what you meant; thank you.

You might, alternatively, have defined the 'radius' of a regular polygon to be its circumradius, i.e. the radius of the circumscribed circle. In the case of a regular polygon with an even number of sides, this would be equal to half its 'diameter'.

As for the missing introductory post: I was just puzzled, because I thought it was technically impossible to begin posting to the other forums until one had posted in the 'New Member Introductions' forum! How did you manage that?

@Reconstructo,

If it appeals to anyone, I would like to steer this thread back to its theme. If a geometry thread were started, I would join it, but this thread is actually about synthesis or dialectic (the triangle.....)

warm regards: recon

Reconstructo;133937 wrote:

The triangle is a great representation of dialectic or synthesis. The circle is a great representation of logic or number.

Pure logic is pure number and pure number is only one number, despite appearances. All number is a modification of 1. Don't be deceived by our ten-digit positional system. It's convenient, as we have ten digits. But ten digits is one quantity. (More on that later...)

Logos, however, is not circular but triangular (or spiral, but that's another thread). For logos, or word, is not only the unification of qualia, but also of other words. Logos is an transcendentally synthetic, but only because it relates us to the incidental, by unifying qualia. The first way that logos is triangularly describe is: one bottom angle is number, and the second bottom angle is qualia. At the top, we have the synthesis of qualia and name, or concept. This is concrete concept, or less-abstract concept.

Sometimes the "triangle" is used to synthesize not qualia but two or more concepts (so the triangle isn't a perfect analogy in this case.) But essentially, the synthesized concepts are the lower corners of the triangle and both are negated/synthesized in the new concept, the one represented by the peak of the triangle.

Philosophy is nothing but this synthesis, by which the transcendental is revealed/abstracted, including the synthetic process itself, which is both transcendental and incidental. (symbolized quite well by Christ...)

The cross (+) expresses the same synthesis, but that's another thread....

@Twirlip,

Twirlip;134316 wrote:

You might, alternatively, have defined the 'radius' of a regular polygon to be its circumradius, i.e. the radius of the circumscribed circle. In the case of a regular polygon with an even number of sides, this would be equal to half its 'diameter'.

There are certainly other ways to do it.

Twirlip;134316 wrote:

I thought it was technically impossible to begin posting to the other forums until one had posted in the 'New Member Introductions' forum! How did you manage that?

It's only impossible for me to start threads.

@Reconstructo,

Be fair. You did write:

Quote:

Would you mind providing the formula? Are we talking about area, sum of the sides? I'm quite interested in this. Any info would be appreciated...

But it did indeed seem like a big digression, and I'm quite glad that it is over - not that I actually know what to say that is on-topic!

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