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Fred Champion

Hijack - from Dividing a Sphere re. Ideal vs Real

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From your post #63:

"I do not have the understanding of solid geometry and calculus it would take to figure if the length of the diamond's side is exactly r, when measured on the surface of the sphere, but it would be "neat" if it was, and would have implications as to what Pi is. That is, as to "why" Pi is. Just one of the things I care about figuring out. If it has already been figured out fine, I am just looking for the answer, one way or the other."

 

Pi is an imaginary number. The result of attempting to describe the imaginary circumference of an object by the length of a line used to scribe that circle. The radius may be a real physical construct; a circle won't be.

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Fred Champion,

 

I don't quite follow.

 

If an ideal smooth sphere is thought about, and a tiny ant who takes steps that are one Planck's length long is to take a straight route away from a point on the surface and stay on the surface, it will arrive at the starting point after a certain amount of real physical steps.

 

Regards, TAR

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The ideal smooth sphere you reference is imaginary. Construct a physical object using uniform building blocks. That object will have volume. Volume is the one and only real physical dimension of real physical objects.

 

You were already on the path to an alternate geometry when you were using spheres to construct objects. I think the sphere is a good choice for the smallest basic physical building block because it seems to be the most efficient way to segregate one volume from another and it is totally uniform. The traditional 3-D (actually 3-plane) geometry of vertices set on orthogonal axes (the centers forming a cube) may be modified by adding one, two, three or four more physical planes. You will find that the tetrahedron forms an entirely satisfactory physical geometry of 7 planes. Note that this is the traditional way of stacking cannon balls.

 

Back to the circle question. Any attempt to construct a real physical circle will produce a polygon. You know that "the area under the curve" in calculus can be gotten by inspection to greater and greater accuracy by counting smaller and smaller squares but even the smallest squares will not align to produce a smooth curve. In the same way, using a uniform physical building block to attempt to construct a physical circle requires both a radius and a circumference of a whole number of those building blocks (or spheres).

 

One interesting consequence of this type of construction is that not every radius will produce a closed polygon. Start with a sphere as the center and lay on spheres at one sphere diameter away from the center. The result is a construct which when measured from the center of each sphere to the center of its neighbor is seen to be a hexagon. I leave it to you to attempt polygon construction with radii of 2 or more.

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Pi is an imaginary number. The result of attempting to describe the imaginary circumference of an object by the length of a line used to scribe that circle. The radius may be a real physical construct; a circle won't be.

 

Nope: pi is an irrational even transcendental number (ie it cannot be represented as either a fraction of two integers nor as a closed algebraic formula) - it has no imaginary component.

The ideal smooth sphere you reference is imaginary. Construct a physical object using uniform building blocks. That object will have volume. Volume is the one and only real physical dimension of real physical objects.

 

Objects have length - really they do; I can measure it, depend upon it, objectively agree it with my colleagues etc. Same applies to area

 

 

You were already on the path to an alternate geometry when you were using spheres to construct objects. I think the sphere is a good choice for the smallest basic physical building block because it seems to be the most efficient way to segregate one volume from another and it is totally uniform. The traditional 3-D (actually 3-plane) geometry of vertices set on orthogonal axes (the centers forming a cube) may be modified by adding one, two, three or four more physical planes. You will find that the tetrahedron forms an entirely satisfactory physical geometry of 7 planes. Note that this is the traditional way of stacking cannon balls.

 

This is pure Euclidean Geometry and not remotely alternative.

Spheres do not actually fill space very well and don't even have an immediately/easily provable densest packing state.

 

What are the 7 planes associated with a Tetrahedron - their are four faces and yet you claim there are another 3 planes intrinsic to the object (another 4 I could understand). * read your followup - a triangular bi-pyramid has 6 faces not 7

 

Traditional Cannon ball stacks are just as likely to be square-based pyramids from memory (at least in the Royal Navy) - larger stacks were triangular prisms

Back to the circle question. Any attempt to construct a real physical circle will produce a polygon. You know that "the area under the curve" in calculus can be gotten by inspection to greater and greater accuracy by counting smaller and smaller squares but even the smallest squares will not align to produce a smooth curve. In the same way, using a uniform physical building block to attempt to construct a physical circle requires both a radius and a circumference of a whole number of those building blocks (or spheres).

 

Calculus is the limit - and is smooth and continuous.

 

I draw a circle in ink on a piece of paper using my compasses - in what way have I needed to use a whole number of building blocks on either the radius or circumference

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Fred Champion,

To further this investigation of yours, I would ask you to look at Janus's rendering in #18 of the 12 segments of the sphere

 

thread.

 

The "cannon balls" when stacked in the 12 segment configuration stack out and fill space as Janus shows. There are intersecting hexagonal planes and intersecting square planes, all built into the configuration.

 

Regards, TAR

 

A combination of triangles and squares with unit sides, that spheres happily accommodate.

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Objects have length - really they do; I can measure it, depend upon it, objectively agree it with my colleagues etc. Same applies to area

Surely Fred's views become understandable when you realise they lack both breadth and depth.

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Surely Fred's views become understandable when you realise they lack both breadth and depth.

 

Brilliant. +1 (actually -1 cos I hit the wrong button - sorry)

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Yes, I should have not have said pi is an imaginary number. I should have just said it is imaginary. I was/am responding to TAR's inquiry into what pi "is" and why it is. I take the "is" in his inquiry to refer to some possible real physical presence and/or significance as opposed to the obvious and well known relationship in the imaginary world of mathematics.

 

Any attempt to construct a circle in the real physical world will require placing real physical objects adjacent to one another (in a line) at a constant distance from a center. Whether these objects are bits of graphite, drops of ink or individual atoms really doesn't matter; they will, as all physical objects do, have volume and occupy that volume to the exclusion of all other objects.

 

Even if the objects thus placed are chosen to be the smallest possible physical objects, when we zoom in on a section of the line we will see the individual shapes do not present a smooth curve. The distance between the centers (or any other consistent place on the shape) will be a straight line (in the case of spheres, one diameter). Those lines will present as a polygon.

 

As Imatfaal posted: "Calculus is the limit - and is smooth and continuous." For TAR's inquiry we must add that calculus is part of the imaginary world of mathematics and the real physical world is not smooth nor is it continuous and remind that objects in the real physical world are discrete things which occupy volume, not imaginary points. Thus there "is" no pi in the real physical world and I hope my explanation of why is adequate.


 

...

 

This is pure Euclidean Geometry and not remotely alternative.

Spheres do not actually fill space very well and don't even have an immediately/easily provable densest packing state.

 

What are the 7 planes associated with a Tetrahedron - their are four faces and yet you claim there are another 3 planes intrinsic to the object (another 4 I could understand). * read your followup - a triangular bi-pyramid has 6 faces not 7

 

...

 

To me the sphere is the closest thing we can get to present a unit of volume and thus are the best thing to use in real world constructs in place of an imaginary point. Is there something better?

 

You're right about the triangular bi-pyramid; it was presented as a reference to visualize two joined tetrahedrons. In a larger volume of many tetrahedrons, the plane where the two tetrahedrons are joined in the triangular bi-pyramid would not be lost.

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Yes, I should have not have said pi is an imaginary number. I should have just said it is imaginary.

 

It is no more and no less "imaginary" than any other number. The Greeks thought that only integers and integer ratios were in some sense "real" but they were obviously wrong.

 

 

To me the sphere is the closest thing we can get to present a unit of volume

 

As pi is irrational, a sphere is a really bad choice as a unit of volume. A cube would be much more sensible. But, as noted, volume is derived from other measurements.

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Yes, I should have not have said pi is an imaginary number. I should have just said it is imaginary. I was/am responding to TAR's inquiry into what pi "is" and why it is. I take the "is" in his inquiry to refer to some possible real physical presence and/or significance as opposed to the obvious and well known relationship in the imaginary world of mathematics.

 

Per Strange - and what ratio is no imaginary to an extent. I don't like the use of imaginary in this context as it both clashes with the non-real components of complex numbers and with the notion of subjective flights of fancy; numbers are non-concrete and abstract, but I would hesitate to call them imaginary

 

Any attempt to construct a circle in the real physical world will require placing real physical objects adjacent to one another (in a line) at a constant distance from a center. Whether these objects are bits of graphite, drops of ink or individual atoms really doesn't matter; they will, as all physical objects do, have volume and occupy that volume to the exclusion of all other objects.

 

No - not at all. I have just flicked the pedal of my bike and the brand new reflecting patch on the back tyre is describing a perfect circle; it's continuous. If we are going to subatomic levels for you to prove graininess - then I would posit the electron orbital; that is pretty continuous.

 

 

As Imatfaal posted: "Calculus is the limit - and is smooth and continuous." For TAR's inquiry we must add that calculus is part of the imaginary world of mathematics and the real physical world is not smooth nor is it continuous and remind that objects in the real physical world are discrete things which occupy volume, not imaginary points. Thus there "is" no pi in the real physical world and I hope my explanation of why is adequate.

 

The path of a flung stone is a continuous parabola (or possibly a hyperbola if you are a pedant), a lightwave is a superposition of two continuous sine waves - sure you cannot weigh out a pound of pi and stick it in a jam jar but that does not imply a chasm between an ideal smooth manifold mathematical world and a lumpen quantised reality. The points you are making are quite clear - and in my opinion totally incorrect. Let us not get bogged down in the ontology of mathematical objects - pi is a very useful tool in the explanation, prediction, and understanding of the physical world; pi is vital in almost every area of physics and crops up almost everywhere; pi, along with e, and i are (for reasons beyond our current understanding) special and interelated to most maths; and whilst pi might seem to be unreal in some lights we have seen enough times in the philopsophy section that extreme solipsism can lead us to that conclusion with anything.

 

You're right about the triangular bi-pyramid; it was presented as a reference to visualize two joined tetrahedrons. In a larger volume of many tetrahedrons, the plane where the two tetrahedrons are joined in the triangular bi-pyramid would not be lost.

 

To an extent as tetrahedrons do not fill space - but I think that any arrangement of tri bi pyramids would be interpreted as a collection of tetrahedrons; we tend to be reductionist when looking a shapes like that

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If we are going to subatomic levels for you to prove graininess - then I would posit the electron orbital; that is pretty continuous.

This is an unfortunate example. The electron orbital is a fiction. Closer to reality are 3D probability, density clouds.

 

I agree with the rest of your argument, but this illustration using orbitals fails.

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...

The path of a flung stone is a continuous parabola (or possibly a hyperbola if you are a pedant), ...

Yay pedants! :lol: Moreover, and pedantically germane to this discussion, spheres in-and-of themselves don't have volume they have area. Balls have volume.

 

It takes balls to be pedantic. :P

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This is an unfortunate example. The electron orbital is a fiction. Closer to reality are 3D probability, density clouds.

 

I agree with the rest of your argument, but this illustration using orbitals fails.

 

Surely that would be an orbit - the atomic and molecular orbitals are the quantum mechanic spatial functions which describe where an electron is likely to be found. It is the orbit (per the SFN logo) which is a fiction - as usual in science we used a word which seemed to be the same to describe a radically different concept.

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And we can make images of orbitals (which match the theoretical images from my old physical chemistry books).

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But none of them, as I recall, are circles. Curves, yes, spheroids, yes, but circles?

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I think the point was more about things being continuous, not made up of straight line segments (rather than circles, per se).

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Fred Champion,

 

 

Yes I find that working with clay things work out and I have not found pi to more than a couple of decimal places. That is , when a soap bubble takes on a spherical shape, there is not a computer required to figure Pi to a million places. There is something about the configuration that is automatic. That continuous thing, the sine waves and smooth curve of a thrown ball, or the path of a rock tied to a string and swung around your head.

 

The circle, does not require Pi to be. It is a ratio of the circumference to the diameter of a circle. There should be a reason for this ratio. That is all I am suggesting. That this ratio means something.

 

Regards, TAR

Edited by tar

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The circle, does not require Pi to be. It is a ratio of the circumference to the diameter of a circle. There should be a reason for this ratio. That is all I am suggesting. That this ratio means something.

Tar; on the subject of dimensionless numbers - Martin Rees the old Astronomer Royal has written a superb book called Just Six Numbers about 6 of the dimensionless (ie with no units such as metres or seconds etc) constants* that seem to govern our universe. pi is not one of his numbers as it is more mathematical in its (unknown) origins - but the other numbers are similarly mystifying. From seemingly the simple question of why 3 dimensions and not another number to questions about why 1/137-ish is of prime important to all matter. Really recommended and asks lots of the same questions you have asked about pi; no answers yet unfortunately for any of these "why" questions.

 

* to explain the "dimensionless" bit. Constants of physics such as Newtons Universal Gravitational Constant have units ie G is in Newtons x Metres Squared / Kilograms Squared. Thus if you change measurement systems (say to imperial from metric) you get a new constant. Some constants are special though and they appear throughout physics or have great significance - yet have no units and thus no matter what measurement system you employ they are always the same. Perhaps the most famous is alpha the fine structure constant (which is close to 1/137) and governs the strength of electromagnetic interaction.

 

pi similarly is dimensionless as it is a ratio of one length (the circumference in metres) to another length ( the radius in metres) and so the lengths cancel and you are left with "just" a number

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Cycloid arcs, while an old concept, provide some interesting insights into circles and [math]\pi[/math], especially when you consider that the area of a complete cycloid plot equals the surface area of a sphere.

 

http://quadrivium.info/MathInt/Notes/Cycloid.pdf

 

In a complete cycloid plot (length [math]2 {\pi} r[/math], height [math]2 r[/math], area [math]4 {\pi} r^2 [/math]) (1) the area above the cycloid curve is equal to the area of the circle that created the cycloid and (2) the area below the cycloid curve is equal to 3 times the area of the circle that created the cycloid.

 

 

 

 

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I think the point was more about things being continuous, not made up of straight line segments (rather than circles, per se).

Yes, I completely get that point and am in agreement with it. I am just uncomfortable with arguments that offer a weak example that an opponent can use to distract attention from the central point. Much as I have inadvertently done here. :)

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Fred Champion,

 

 

Yes I find that working with clay things work out and I have not found pi to more than a couple of decimal places. That is , when a soap bubble takes on a spherical shape, there is not a computer required to figure Pi to a million places. There is something about the configuration that is automatic. That continuous thing, the sine waves and smooth curve of a thrown ball, or the path of a rock tied to a string and swung around your head.

 

The circle, does not require Pi to be. It is a ratio of the circumference to the diameter of a circle. There should be a reason for this ratio. That is all I am suggesting. That this ratio means something.

 

Regards, TAR

Well of course the ratio we call pi means something. The meaning is defined as a mathematical ratio, not a physical ratio. There is no physical meaning. Circles, arcs, paths, sine waves, other sorts of smooth (continuous) curves and straight lines, and ratios are mathematical concepts, not physical entities.

 

Mathematical constructions may take on whatever form is consistent with the math system used. Physical constructions are consistent with the current state of the physical environment. The most widely adopted math uses continuous concepts to describe and approximate (to a useful degree) observed physical conditions and behavior. Physical constructs are not continuous and are in no way dependent upon any mathematical description.

 

Mathematics doesn't "do" anything. If we write every known symbol and equation in every configuration we can think of on a blackboard and then sit back and contemplate our efforts, what will happen? No amount of conjuring or number of incantations will bring forth activity from all that "math". The chalk may eventually lose adhesion and drop off and the blackboard may eventually crumble onto the floor but the math will have had no effect on anything. There is no magic in it.

...

From seemingly the simple question of why 3 dimensions and not another number to questions about why 1/137-ish is of prime important to all matter. Really recommended and asks lots of the same questions you have asked about pi; no answers yet unfortunately for any of these "why" questions.

...

So many people seem to be absolutely convinced that the universe really is physically 3-D. Not so. 3-D is actually three plane geometry. The universe has only one physical dimension which is what we call volume. Physical volume cannot be reduced to physical "area" nor to physical "length". Area and length are wonderful and useful concepts but are not physical things.

 

Why a geometry of 3 planes or 3 "dimensions"? Because 3 orthogonal planes is the minimum number required to describe the physical dimension of volume, and is the simplest to visualize.

 

Most of us have studied (I expect) other systems of geometry such as spherical and cylindrical. I have reminded that it is possible to set up 4, 5, 6 and 7 plane geometries in addition to the common 3-D. The 7 plane is more interesting because it is symmetrical.

 

"Why" is usually not a good question; we usually can't get a satisfactory answer.

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Fred Champion,

 

There is a philosophical, or metaphysical component to deciding what is mathematical and what is physical. Kant describes language in such a way as to consider reality as being something in and of itself, that is different from what we can say about it. Questioning as to whether or not we know the thing as it is, by merely making a judgement about the thing, enough to say something about it.

 

My personal way to parse this situation is to consider that ALL of my perceptions and judgements are like the shadows on the wall of Plato's cave. In this light, we each have a model of the world built in the synapses and structure and connections and positions and motions in our brains and body and heart. And to the extent that math is a language, math is simply something that we can say about the world. Like you say, all the math in the world. written on the chalkboard will not "do" anything. A mathematical description of the exact location and motion of every quark in a peanut butter cup, will not taste sweet, creamy, nutty and chocolaty.

 

But when I make a sphere out of clay the ratio of its diameter to its circumference is always Pi. Even when I don't have a calculator to know what that number is. I don't have to say anything about the ratio, it is already there. It already exists as a thing in itself. No matter how big the ball, if I measure the things diameter and I roll it on a table and measure the distance on the table where the same point touches again after rolling it in a straight line, the two distances are at the same ratio to each other. Always Pi.

 

Therefore Pi is a physical reality. The breakdown of Pi, or the figuring of it based on polygons and straight lines and such is the mathematical part, in terms of what we can say about the relationship or how to describe it to each other in common language. But a smooth curve, like you would get if you followed the path of a stone on the end of a string swung around your head, is a smooth curve. Really it is. It is not made up of integrals of arbitrary size, except in analogy taken in our heads. It is actually. physically a smooth curve.

 

Regards, TAR

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So many people seem to be absolutely convinced that the universe really is physically 3-D. Not so. 3-D is actually three plane geometry. The universe has only one physical dimension which is what we call volume. Physical volume cannot be reduced to physical "area" nor to physical "length". Area and length are wonderful and useful concepts but are not physical things.

 

Nonsense. Our perceived universe is three dimensional (there may be other per some of the string theories). There are lots of coordinate systems which we can use - claiming that we need to use three planes or that that IS the correct method of description is just madness - these are methods for determining positions when all is said and done and you use the most convenient. Volume is an abstraction just like area and length - that you privilege one over the others is your choice - but don't be fooled into thinking you are correct is this.

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Perhaps the best analogy to determine what is real and what is a number describing reality is to consider analog to digital conversions. The analog voltage really exists. The digital representation of that voltage, as a base 256 number is math. The analog voltage is the physical reality.

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