Mad science: changing pi

[Telok]

Honesty: its been too many years since schooling and I'm not sure my math was quite up to this even then.

Assume you have a box. The box is sufficently advanced tech from aliens or something. There is a dial on the box, and a window I guess. Within the box, by turning the dial, you can change the value of pi. We put things like marbles, rubber balls, and cups of tea in the box. Then start turning the dial.

Now pi is just a value derived from how we calculate the shapes of stuff, specifically circles and. So what has to be happening (sufficently advanced) is, I think, the box janks space-time or something. Euclidian geometery goes "phbttt".

From doodling, around if you push pi to 4 a radius 1 circle goes square-oid-ish, with sides of 2, area 4, circum of 8, and all points on the edge 1 from the center, including the corners. Probably hurts the eyes to look at. Push pi to 5 and... almost a triangle-oid-ish? Cut pi to 2 and... Well hit my time limit anyways. Feh, google, looks like maybe it turns into a line? But that screws up the area.

That sound somewhere near right-oid-ish?

[NichG]

I had a professor once who described certain things as having 'units of pi'. It would definitely make this easier to answer if we actually tracked angular units throughout every physical constant and formula.

For example, Coulomb's Law can be expressed as having a 4pi in the denominator, which suggests units of solid angle. But in that form there's a physical constant 'the permittivity of free space' e0. Even though the origin of the formula is basically the same, the standard form of Newtownian gravity isn't expressed with a 4pi. So that makes me think that e0 would also probably change if you changed pi, keeping the force between a pair of particles of fixed charges separated by a fixed distance the same.

But maybe not! If there's 'more space per unit distance' to space, then fields should fall off more quickly. In which case, maybe when you turn the knob stuff explodes because now all the chemical bonds are the wrong length and that's a lot of energy to dump into every part of a piece of matter at a given time.

If only we had kept track of those units of pi...

[halfeye]

I had a professor once who described certain things as having 'units of pi'. It would definitely make this easier to answer if we actually tracked angular units throughout every physical constant and formula.

For example, Coulomb's Law can be expressed as having a 4pi in the denominator, which suggests units of solid angle. But in that form there's a physical constant 'the permittivity of free space' e0. Even though the origin of the formula is basically the same, the standard form of Newtownian gravity isn't expressed with a 4pi. So that makes me think that e0 would also probably change if you changed pi, keeping the force between a pair of particles of fixed charges separated by a fixed distance the same.

But maybe not! If there's 'more space per unit distance' to space, then fields should fall off more quickly. In which case, maybe when you turn the knob stuff explodes because now all the chemical bonds are the wrong length and that's a lot of energy to dump into every part of a piece of matter at a given time.

If only we had kept track of those units of pi...

Aren't those radians?

[NichG]

Aren't those radians?

Radians for angles, steradians for solid angles. But you don't usually see those kept around as units in fundamental constants.

For example, h and hbar are written as having the same units, but differ by a factor of 2pi. But if you change pi, is h constant, or hbar?

[Seppl]

Mathematically you can get any value for pi between 2 and 4 by just changing what metric you use to measure distances. The choice of Euclidean metric as a default in geometry is no accident (it is rather convenient and intuitive) but that choice is not necessarily grounded on physical reality. For all we know, reality might as well be "pixelated" and taxi-driver distance (where pi=4) might be the more "natural" metric of real space. We even know that the laws of gravity can be expressed in way simpler form in terms of curved space-time than if we assumed flat Euclidean space. What I am saying is that both Pi and Euclidean geometry are both theoretical concepts, with no base in reality. They are both useful choices for talking about the universe in broad strokes on length scales that are interesting to humans, but the universe in general does not care which equations we chose to describe it.

[Chronos]

Are we defining pi as the ratio of a circle's circumference to its diameter, or a circle's area divided by the square of its radius, or the surface area of a sphere divided by 4r^2, or the volume of a sphere divided by 4/3 r^3? Or maybe something weirder, like sqrt(6*(1/2)!) ? All of those would have different consequences.

[Quizatzhaderac]

From doodling, around if you push pi to 4 a radius 1 circle goes square-oid-ish, with sides of 2, area 4, circum of 8, and all points on the edge 1 from the center, including the corners. Probably hurts the eyes to look at. Push pi to 5 and... almost a triangle-oid-ish? Cut pi to 2 and... Well hit my time limit anyways. Feh, google, looks like maybe it turns into a line? But that screws up the area.

That sound somewhere near right-oid-ish?

There are different possible visiual, depending on exactly how the warped space in the box related to Euclidean space outside.

One case where pi is less that 3.14 is on the surface of a sphere (such as the earth). If one was the draw a 10,000 Km line from the north pole to the equator and call that line the radius, and the equator the circumference, you would find that pi equals 2. However, looking from space at the north pole, that 10,000 m radius line would look less than 10,000 Km, because it's curving away from the viewer. BY changing your perspective, the equator line may end up looking like an ellipse.

A case where pi would be more than 3.15 would be if you drew a circle on a saddle. The circumference line would have to go up, down, up, and down again, so the circumference line would be much longer than if it was on a flat surface. But when looking at it from straight above, the circumference line would still only appear to be 2 pi times what the radius line appears to be. From an angle the circle would appear to be, I don't think there's a proper name, so I'll call it a wibbly-wobbly ellipse.

[Radar]

There are different possible visiual, depending on exactly how the warped space in the box related to Euclidean space outside.

One case where pi is less that 3.14 is on the surface of a sphere (such as the earth). If one was the draw a 10,000 Km line from the north pole to the equator and call that line the radius, and the equator the circumference, you would find that pi equals 2. However, looking from space at the north pole, that 10,000 m radius line would look less than 10,000 Km, because it's curving away from the viewer. BY changing your perspective, the equator line may end up looking like an ellipse.

A case where pi would be more than 3.15 would be if you drew a circle on a saddle. The circumference line would have to go up, down, up, and down again, so the circumference line would be much longer than if it was on a flat surface. But when looking at it from straight above, the circumference line would still only appear to be 2 pi times what the radius line appears to be. From an angle the circle would appear to be, I don't think there's a proper name, so I'll call it a wibbly-wobbly ellipse.

In the examples above you are specifically working with pi defined as the ratio between circumference of a circle and its diameter, which is just one of the options laid down by Chronos. Regular constant curvature, locally Euclidean spaces will also have a specific problem: said ratio will not be a constant, which is what OP wanted to explore.

Regardless of which definition of pi you take, making it a constant different than 3.14... would require a metric that even locally is very much non-Euclidean. Even with that I am not sure if it is possible to have for example pi*r^2 as the circle's surface and 4pi^2*r^2 as the sphere's surface at the same time, so it is important to state, which relation containing pi you want to directly tweak as the others will follow but will be modified in a different way.

[Telok]

Um, honestly, the whole thing came about from an idea for a space opera super weapon that made a zone where an effect made pi go off and the pipes/ ball bearings/ round things all suddenly had weird gaps/ overlaps /something to make the target building/ vehicle/ person go... and then I wasn't sure if it would be "splort" "boom", sudden existance failure, rapid uncontrolled disassembly, or what.

I know you could match it all up in non-Euclidian fooling in higher dimensions. Like... like a 2d flatlander machine with gears & rods getting partially onto a curved bit of paper? So for us if the axle & wheel of a car has chunks go all dimensional on us it looks like ??? and results in ???. That was sort of the thought process.

[Anymage]

Real physics, if you can warp spacetime like that, you definitionally have the ability to weaponize gravity. You can cause a lot more damage than just causing round things to be the wrong sort of round.

Space opera, I'd say that you get a nice visual by essentially lowering a round thing's poly count. Things can look weird by fitting together in ways they shouldn't, so long as they're stationary. Things in motion can fall out of place, meaning that life processes start to feel uncomfortable and any sort of gross motion (including moving limbs as well as any mechanical objects) will cause things to fall out of alignment. That's more based on a narrative sense of "what if you weaponized R'lyeh's geometry" than trying to work out especially bizarre curved space metrics, however.

[The Glyphstone]

That, or replicating Bloody Stupid Johnson's inventions.

[Radar]

That, or replicating Bloody Stupid Johnson's inventions.

Or something inspired by the improbability drive just without a normality restoration mechanism. Come to think of it, Dr. Strange has very fitting visuals for this kind of reality-warping effect.

Narratively, something alike to Lovecraftian geometry would be very fitting as it was more or less meant to be what feeble human minds got out of seeing higher-dimensional beings.

[Chronos]

Come to think of it, yet another one: You can define pi as half the period of a function that is the negative of its own second derivative.

[NichG]

Come to think of it, yet another one: You can define pi as half the period of a function that is the negative of its own second derivative.

Now I want to see the algebra constructed around a field whose numbers allow this property to vary. I suppose it's natural to start with the complex numbers since that has a relationship between derivatives and values if you're working with harmonic functions... And it also connects products to rotations.

So maybe something like extended complex numbers, where instead of representing a 90 degree rotation, i represents either a greater or lesser angle of rotation? So instead of i^4 = 1, you have i^(4+alpha) = 1, with alpha indicating how detuned things are from Euclidean. Then keep all the complex derivative/harmonic function stuff the same, and see what you get when you write down f(z)'' = -f(z) in that space?

Maybe this is overcomplicating it and if you just have a curved metric over x, its enough. Like solving the wave equation in flat space gives you sines and cosines, but the same equation on a sphere gives you spherical harmonics.

[Trafalgar]

Honesty: its been too many years since schooling and I'm not sure my math was quite up to this even then.

Assume you have a box. The box is sufficently advanced tech from aliens or something. There is a dial on the box, and a window I guess. Within the box, by turning the dial, you can change the value of pi. We put things like marbles, rubber balls, and cups of tea in the box. Then start turning the dial.

Now pi is just a value derived from how we calculate the shapes of stuff, specifically circles and. So what has to be happening (sufficently advanced) is, I think, the box janks space-time or something. Euclidian geometery goes "phbttt".

From doodling, around if you push pi to 4 a radius 1 circle goes square-oid-ish, with sides of 2, area 4, circum of 8, and all points on the edge 1 from the center, including the corners. Probably hurts the eyes to look at. Push pi to 5 and... almost a triangle-oid-ish? Cut pi to 2 and... Well hit my time limit anyways. Feh, google, looks like maybe it turns into a line? But that screws up the area.

That sound somewhere near right-oid-ish?

I actually once read a book where this occurred. It stated that, several thousand years ago, the value of pi was 3.0. Sometime in the intervening time, the value had magically been changed to 3.14. I can't remember what shape the item was in the book, I think it was still described as "round".

[NichG]

Following on the last thing about the algebra of these different-pi spaces, I think we might actually need to redefine the multiplication operator even for things like positive/negative real numbers...

So lets say we have a space of numbers formatted a*u + b*v where a and b work like real numbers and u and v are unit vectors sort of like how i works, or i,j,k for quaternions. But now you can't have a 'bare' number in the system like a + b*v.

Now, if we define how u and v unit vectors interact, we have a whole set of number spaces. E.g. u*u = alpha * u + beta * v, v*v = gamma * u + delta * v, u*v = epsilon * u + rho * v

So now if you define a derivative in this space via a limit, you have e.g.:

df/du = lim[delta->0] (f(x+delta u) - f(x)) / (delta u)

So the 'u' part of f goes as:

df/du = alpha^-1 * df/dx * u + beta^-1 * df/dx * v

where df/dx would be the real number equivalent.

So that could easily let you 'change pi' even in the sense of pi = half period of f'' = -f.

[Radar]

Now I want to see the algebra constructed around a field whose numbers allow this property to vary. I suppose it's natural to start with the complex numbers since that has a relationship between derivatives and values if you're working with harmonic functions... And it also connects products to rotations.

So maybe something like extended complex numbers, where instead of representing a 90 degree rotation, i represents either a greater or lesser angle of rotation? So instead of i^4 = 1, you have i^(4+alpha) = 1, with alpha indicating how detuned things are from Euclidean. Then keep all the complex derivative/harmonic function stuff the same, and see what you get when you write down f(z)'' = -f(z) in that space?

Maybe this is overcomplicating it and if you just have a curved metric over x, its enough. Like solving the wave equation in flat space gives you sines and cosines, but the same equation on a sphere gives you spherical harmonics.

This seems to create a complex space spanned on a cone (with 0 at the pointy top) or whatever the hyperbolic counterpart would be. So you would have a non-differentiable point and a fairly well understood curvilinear space everywhere else. Would require a different definition of the scalar product though as I*(-I) would not be real, but in non-Euclidean spaces that is pretty standard.

Interestingly, there are ways to make weirdly long circles in complex spaces that do not need any redefinition of I. Any multivalued complex function contains specific points (either with value 0 or a singularity) around which the full angle (as in the ratio between a circle's circumference and diameter) is a multiple of 2 pi. For example, simple square root is one of those functions. Any monomial with non-integer power will induce such a space with the number of branches you need to loop around depending on the denominator in the power. Typically z^something irrational is considered to have infinitely many branches that do not loop around, but if you force things to fit onto a single branch, there would some serious shenanigans needed with the arguments as we would have to skip some part of the complex plane or repeat just a part of it in order to connect the values smoothly. How to define such an operation, I do not know to be honest.

Hmm... maybe something weird can be done with modulo rings over real or complex numbers?

[Bohandas]

I think this would be akin to changing the curvature of the space inside of the box. In elliptic space a circle's circumference can become arbitrarily small in comparison to its radius, and I'm pretty sure in hyperbolic/lobachevskian space I'm pretty sure it can become arbitrarily large

[Radar]

I think this would be akin to changing the curvature of the space inside of the box. In elliptic space a circle's circumference can become arbitrarily small in comparison to its radius, and I'm pretty sure in hyperbolic/lobachevskian space I'm pretty sure it can become arbitrarily large

Overall yes, just that the ratio between circumference and radius will scale with size. Making pi have a constant value different than that for Euclidean space is far more tricky. Not necessarily what OP needs, but it is a fun problem to tinker with.

[Bohandas]

Overall yes, just that the ratio between circumference and radius will scale with size. Making pi have a constant value different than that for Euclidean space is far more tricky. Not necessarily what OP needs, but it is a fun problem to tinker with.

You'd need to balance it with some sort of gravity-like effect that scales with the object's size instead of its mass

[Radar]

You'd need to balance it with some sort of gravity-like effect that scales with the object's size instead of its mass

On the level of physics it's one thing, but the main challenge is to establish any kind of geometry that works like this. Basically, the metric would have to be very different from what we are used to. For example, taxicab metric (distance on a plane is counted as the sum of distance in X and Y directions) would result in pi=4 regardless of the size of a circle, but it does not even have uniquely defined straight lines (shortest lines between two points) and is not rotationally invariant.

[Telok]

I may or may mot have comitted a terrible crime. In an obscure bit of world building I wrote "pi elemental".

Time, gravity, plasma, and cheese elementals are easy*. I'm not sure what happens when a pi elrmental hits someone. Geometry elemental even, sure. You get rearranged to look like you got shoved through a drunk cubist painting and came out the worse for wear. But a pi elemental?

* time will age to death or reverse to infancy, gravity just crushes or flings you half a mile, plasma is burns, cheese is just cheddar or maybe nausea if its an overripe limburger unless you're lactose intolerant then its save or swell up and die.

[NichG]

Well, when pi 'hits' something (e.g. multiplies it), it turns linear things into angular things. So you get bent into a donut.

[Radar]

Well, when pi 'hits' something (e.g. multiplies it), it turns linear things into angular things. So you get bent into a donut.

Well, the effects of being hit by pi are well understood.

Aside from that, depending on what kind of world building material that was written in, there is always an option of unreliable narrator and the actual monster can be almost anything. Narrative-wise an interesting thing would be to consider that some mad wizard or god created a semi-elemental plane of pure mathematical concepts just to prove a point in a philosophical debate, but did not pay much attention to the details.

[Khedrac]

To add to the fun, somewhere there is a short story by Colin Kapp (I think it was "Getaway from Getawehi") where a bunch of engineers were trying to cope with a planet so far out from "normal" regions that 1 + 1 ≠ 2 (I think it was about 1.8), This manifested in that if you took two metal rods of equal length, and then cut one in half and re-joined it, it would now be considerably shorter than the other one. Imagine trying to assemble kits in such conditions...

As for changing pi, something that hasn't been mentioned is that regardless of what would happen in the area, I am not sure you would be able to see it - if space-time is playing games and falling apart then light isn't necessarily going to reflect off surfaces and make it back out of the zone for us to see what happens

[Rockphed]

Even with that I am not sure if it is possible to have for example pi*r^2 as the circle's surface and 4pi^2*r^2 as the sphere's surface at the same time, so it is important to state, which relation containing pi you want to directly tweak as the others will follow but will be modified in a different way.

Surface area of a sphere is 4pi*r^2. I cannot think of anything off the top of my head where pi gets squared.

[NichG]

Surface area of a sphere is 4pi*r^2. I cannot think of anything off the top of my head where pi gets squared.

There's some infinite sums that resolve to pi^2 (but of course, by definition, thats true of basically any continuous function of pi as well due to Taylor series...). These sums evidently show up in some Feynman diagram stuff, but I have no geometric or physical intuition about them.

On the other hand, sqrt(pi) shows up in the normalizing factor of a Gaussian, which might be easier to come up with a geometric sense for?

[Radar]

Surface area of a sphere is 4pi*r^2. I cannot think of anything off the top of my head where pi gets squared.

Yeah, one can reread the comment before posting and still leave such an error. The point still stands that with changing geometry, both relations will change in a different way.

[Bohandas]

Surface area of a sphere is 4pi*r^2. I cannot think of anything off the top of my head where pi gets squared.

I think it happens in higher dimensional geometry

edit:
https://en.wikipedia.org/wiki/Volume_of_an_n-ball

edit:
https://en.wikipedia.org/wiki/N-sphere

[Rockphed]

I'm not even going to try to pretend how things work in spaces with more than 3 dimensions. When I use more than 3 dimensions it doesn't correspond to a physical space, just sets of measurements. That said, I have some experience doing things in 1-, 2-, and 3-dimensions that I think is relevant.

In a 1-dimensional space, conservation of energy means that a traveling wave always maintains constant energy. In a 2-dimensional space, conservation of energy leads to the energy being spread around the circumference of a circle and falling off as E/(2pi*r). In a 3-dimensional space, this happens as E/(4pi*r²). Pi, in this case, is a ratio between how far something goes and how much it gets spread out. So if we increase pi then energy fields falls off faster as they travel and if we decrease pi they fall off slower; gravity and electromagnetism should have an inverse strength relative to changes in pi.

On the other hand, somebody mentioned complex numbers and periodic functions. The most basic being e^(i*x), with the solution of e^(i*pi) = -1. If we increase pi it creates more space for waves to travel through in a period, which means that everything would be red shifted.

This leads to an interesting question: is "Dark Energy" just the result of pi having increased over the history of the universe?