Naked Science Forum
General Science => General Science => Topic started by: jeffreyH on 06/02/2018 21:24:18
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Since a sphere has a bounding surface is it possible to have a sphere with infinite radius? If not then is it possible to have an unbounded space?
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Well...
There is no limit to how large a sphere can be (in theory). For any sphere of finite size, there are infinitely many possible spheres that are larger.
How you define a sphere then depends on whether there can be an infinitely sized sphere. For instance I think there wouldn't be any major problem defining some point as the origin of the sphere, and a locus of points all the same infinite distance away as the sphere (not very useful though). But I think that there would be many problems at the "surface" of a sphere with infinite radius. The local curvature would be related to the radius--and in the case of an infinite radius, I don't think there is any characteristic that would distinguish this case from a perfectly flat plane.
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The Earths surface has no boundary, but is only 2 dimensional. So a hypersphere's surface would also be unbounded since it would have no boundary, while being 3 dimensional, but be a finite volume.
It's thought the universe might be aproximately hyperspherical.
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It's interesting that you equate the surface of the sphere with a flat plane. Would you consider this to be a plane that is infinite?
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No, but it's unbounded.
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If the universe has a finite radius, then it can't contain a sphere of infinite radius.
However, if you include all points within the radius of the universe as "inside" the sphere, then there are no accessible points "outside" the sphere.
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Think about this:
Consider an n-sphere. For any natural number, n, the n-sphere is defined to be embedded in a Euclidean (n+1)-sphere, where r ∈ +Reals. This would then imply that a sphere of infinite radius is possible in (n+1)-dimensional Euclidean space, and might be defined as:
Sn={x∈Rn+1:||x||=r}
There is one limiting factor here, though. As the radius increases, the curvature of the sphere decreases. As the curvature decreases it approaches the 0-curvature of a hyperplane. However, considering the asymptotic nature of this restriction, as r → infinity, n-sphere → hyperplane should be an impossibility.
I think. :)
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A sphere is the locus of all points at radius r from the centre.
However large you make r, I can conceive of a sphere of r + Δr or even 2r, so you can't conceive of, let alone make, an infinite sphere.
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However large you make r, I can conceive of a sphere of r + Δr or even 2r, so you can't conceive of, let alone make, an infinite sphere.
Hi and Thank you for continuing the conversation.
I mean no disrespect, but the above statement seems to contradict itself. If we create a sphere of radius r, then obviously any sphere of radius r+n would be possible. Wouldn't a sphere of radius r+n, or just n, therefore be considered a sphere of infinite radius? Help me out. I admit I may be missing something.
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No, because I can conceive of a radius r +n +1. Infinite means "nothing can be larger than....". Merely running out of bricks doesn't mean you have built an infinite wall because, as the Panzer Divisions discovered, you can drive round it with a finite amount of fuel.
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If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.
There would not be enough room in an infinite universe, for a sphere of infinite radius. We could only fit about an eight of that sphere. The rest of the sphere would need to extend into other dimensions and would need seven other alternate infinite universes in a 2x2x2 configuration.
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An interesting argument is made by puppypower. However, I must respectfully disagree.
A scaling factor, such as 2, would not change the limitless character of infinity.
2∞ = ∞ + ∞ = ∞
Nor would an infinite power.
2∞ = ∞
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I'm curious to learn how to measure an eighth of infinity. @puppypower Care to elaborate?
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If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.
You have obviously never heard of positive and negative infinity. You need to brush up on your mathematics. Try studying limits.
https://en.m.wikipedia.org/wiki/Limit_(mathematics) (https://en.m.wikipedia.org/wiki/Limit_(mathematics))
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I'm curious to learn how to measure an eighth of infinity. @puppypower Care to elaborate?
I used the concept of multi-dimensions and multi-universes, to extend infinity beyond the limits of just one infinite universe. The final shape, using eight universes; 2x2x2, will look like a single sphere of infinite diameter (not radius) that will be overlap all the eight universes. The affect of the infinite radius paradox, is felt by all; synchronized.
As an analogy, say I need to measure a broom handle that is several meters long, but in a single measurement. However, all I have is a 1 foot ruler. I may need to ask my friends to borrow their 1 foot rulers, so I can tape them all together until I get the length needed. Since all the rulers are needed to measure the broom, the length of the broom is a function of a multi-ruler collaboration.
This collaboration is not dependent on which ruler comes first, second or last. It only requires that they all are connected in a line. In the case of the universe, we end up with a type of holographic sphere of infinite diameter (not infinite radius) common to all, but not at all points, since we each have a tangible eighth of a sphere and 7/8 that is not as easy to measure, yet there is impact.
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It is not sufficient to cherry pick scientific sounding words and string them together inside nonsense. It is much better to learn what the words mean. That way you can participate in intelligent debate. I would rather be having such debates with you than largely ignoring your contributions. It is not a weakness to be ignorant on a subject. It is an opportunity to learn. Learning can be very rewarding.
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If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.
You have obviously never heard of positive and negative infinity. You need to brush up on your mathematics. Try studying limits.
https://en.m.wikipedia.org/wiki/Limit_(mathematics) (https://en.m.wikipedia.org/wiki/Limit_(mathematics))
Say we use negative and positive infinity. We start at the origin, and begin with an infinite line to define our radius. I will define this in the (1,1,1) direction. This allows the radius to fit in an infinite universe. Since we are forming a sphere; locus of points equidistant from a single point, we need to move that radius in two angles, to cover the volume of the sphere. To do this, we will need to enter negative infinity territory on the other side of the origin. (-x,-y,-z)
If we fill this sphere in, we end up with as much negative and positive infinity. Relative to our diameter, since diameter in all directions is r plus -r, this equals zero. It cancels itself out. It does not exist. To make it exist, we need to think outside a single universe box of negative and plus, to avoid plus and minus infinity cancelling. That is plus and minus work in math.
I have the mind of a contriver or engineer, so I am wired to figure out to make it work, instead of surrender to the limitations of traditions. I could not make it work, when negative cancels positive. This was considered. There was not enough room for a diameter of 2 times infinity in any positive only infinity quadrant. However, I could make it work using all positive, but that required only an 1/8 of the sphere, in each of 8 overlapping universes. The final shape is an infinite diameter universe hologram, in each of the 8 stations. The sacred nature of infinity is preserved. It is like the speed of light does not change.
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It is not sufficient to cherry pick scientific sounding words and string them together inside nonsense. It is much better to learn what the words mean. That way you can participate in intelligent debate. I would rather be having such debates with you than largely ignoring your contributions. It is not a weakness to be ignorant on a subject. It is an opportunity to learn. Learning can be very rewarding.
I agree with you. However, sometimes what is already there, cannot get the job done, without getting too confusing. This is when you may need to step outside the box. Outside the box is often harder to describe, since there is no tradition for a direct translation. The bridge often has to use common language, but with a slightly new definition added.
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The number line can be thought of as ranging from minus infinity to positive infinity. You don't subtract the negative part from the positive part. It is a continuum and essentially unbounded.
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Hello, again...
There are so many ways to approach the definition of the infinite, yet those ways will all bring us to the same conclusion. As a species we tend to want to quantify everything. It is generally difficult for us to truly rely upon a purely conceptualized reality. It is pointless. We must either agree, or agree to disagree. I am reminded of a book by Ralph Cudworth in the 17th century where he poses "how many angles can dance on the tip of needle." Who cares?
Now, that is not to say that a debate about the scale and scope of infinity is pointless. Infinity is a defined mathematical construct. It is limitless. Anyone who has an understanding of Lobachevskian geometry, limits, or even asymptotic functions has already accepted the boundless nature of infinity. What is, in my opinion, pointless is attempting to quantify infinity. By including any scaling factor in any discussion centered on infinity, such as 2∞, ¾∞, √∞, or even ∞x, we are attempting the impossible. Mathematics is, after all, a "religion." In order for it to work certain assumptions, or axioms, must be accepted.
For example: 1 + 1 = 2 Now prove it.
Wait a moment, before you throw me on the trash heap of the inane, allow me to clarify...
In order to prove 1 + 1 = 2 it must first be proven that more than one "1" can exist. You cannot simply say "I have one orange here, and over there I have one orange, and that makes two "1"s. The moment you identify having two seperate 1s you have "2". So, it must be accepted that there can exist, at least, two distinct 1s. Then you need to prove the concept of being able to "add" them together. Not so simple an idea. Why? Allow me to continue... If I had two distinct containers of water, and then I poured both of them into a single empty container, (adding them together), do I still have two distinct containers of water? No. I have something entirely different. But if I had two distinct containers where one held water, and the other say - milk, and I "added" them in a third container, do I still have two distinct containers simply mixed together? Yes. It would be possible to "filter" the milk from the water.
My labored point is, and please forgive me for that labor, that we need to accept many things as being so, if we are to contemplate what could be. I accept the concept of infinity. I understand it. It is boundless, limitless, dare I say "godly". (little "g", not big "G".) Once I accepted it, the fantastic scope of mathematics was unveiled. Imagine where we would be today if Zeno of Elea had been able to accept the idea of a limit, 2100 years before Leibnitz? Wow!!!
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Mathematics is not religion. You do not have to take mathematics on faith. It is mathematics where proofs are found. Now go and prove a religion in the same way. You might as well punch smoke.
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https://www.bestphotosworld.com/beautiful-smoke-art-photography-examples/amp/
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JeffreyH, based upon your "conversation" thus far, I have nothing but respect for you opinion. I am sorry if I struck a nerve, so to say.
Mathematics is, after all, a "religion."
My characterizing mathematics as a "religion", and note the quotations, was merely to acknowledge that there are some things we have to accept axiomatically. That is, accepted even in the absence of proof. That is all.
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@petelamana Apologies for the terse reply. I found your post very interesting.
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@petelamanaI found your post very interesting.
Which is more than can be said for puppy’s ramblings. He still hasn’t explained 1/8 of infinity, im tempted to move his posts to new theories rather than clogging up bandwidth here.
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For those interested in such things, it is worth noting that the volume created by rotating a hyperbola around its asymptote is finite, but the surface area is infinite. This is of enormous importance to the paint industry: you can cover an infinite area of, say, steel sheet, with a gallon of paint, then cut it up and make, say, a red car, or even a million red cars, with just one gallon of paint!
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Consider an n-sphere
One of the hypotheses on which string theory is based is that the universe has 10 dimensions, but we don't see most of them because they are "rolled up" very tightly (much smaller than the radius of a proton).
So in these (hypothetical) dimensions, the n-sphere (or n-ellipsoid) would have a quite finite and very small radius.
See: https://en.wikipedia.org/wiki/String_theory#Extra_dimensions
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Mathematics is not religion. You do not have to take mathematics on faith. It is mathematics where proofs are found. Now go and prove a religion in the same way. You might as well punch smoke.
Actually there is one very fundamental mathematical operation that needs to be taken with faith. This is division by fractions. If I have one gallon of gas and divide that by 1/10, I get 10 gallons of gas. This operation violates energy conservation since it adds gas out of nowhere. If this violation of energy conservation was valid, then the operation would need to be a magic trick, with a hidden compartment, so we can add the extra nine gallons, when nobody is paying attention.
Let me look at this closer; If I have one gallon of gas and divide it by 10, I separate that one gallon into 10 equal parts, each with 1/10 gallon. This can be done in reality and makes perfect sense. If I apply this division logic to the one gallon divided by an 1/10, I separate the one gallon into 1/10 equal parts, to get a new total of 10 gallons? How is that done and how does it avoid violating energy conservation? Could it be done in the lab? Would dividing a gallon of high and low octane gas by a 1/10 takes different amounts of energy? The operation is based on a procedure that needs faith and cannot be done in the lab. It will have the same success, as making God appear in the lab.
The Lorenz factor in Special Relativity uses this faith procedure. The procedure, itself, could create devotion. The real point I am making is convention does not always have to make sense, if we all go along. Math does not have to reflect the limits of reality, if we all agree to ignore the shortcomings. Imagine if science required all math operators and assumptions run a tangible experiment, to show the operation, by itself, can be done in reality.
This is subtle, but this magic in math is useful and is needed to escape reality, into the world of human imagination so we can make synthetics and improve on nature. A sky scraper is not natural but is helped by math.
@petelamanaI found your post very interesting.
Which is more than can be said for puppy’s ramblings. He still hasn’t explained 1/8 of infinity, im tempted to move his posts to new theories rather than clogging up bandwidth here.
The concept of infinity was original defined, based on a universe perspective of a singular universe. To solve the problem at hand, I needed to change the limits of infinity, to reflect the newer premises in science. I worked under the assumption there are multi-universes. These others universes, are not side-by-side, defining part of original infinity in our universe. I assume they all overlap, in their own dimension, where each universe is unconscious of the others and therefore each can be used to define infinity, based on itself. Each universe is self absorbed, and sets conventions for infinity as though it is all there is; all it can see. We can't see the other universes, so each assumes they are the matrix by which infinity has it limits. Again, when our infinity convention was first cast into some, this applied to our own self absorbed universe.
The analogy that I saw is like layers in Photoshop. With layers in Photoshop, each layer can define an entire picture by itself. No layer is a fraction of a full picture, since all layers are the same size. These multi-universe layers are all stacked, each independent of the other. They can also be blended, so they all superimpose in part, or whole.
Since infinity in each layer is the limit convention, seen by each layer, a radius of infinity already extends to the limits in each layer. Therefore, it is not possible to define a sphere of diameter of 2r in any one layer, since that would mean double infinity, which is a contradiction in terms of the convention in each layer. The story should end there.
However, since I have many layers, with the inhabitants of each layers, self absorbed and assuming they can hold a full infinity, I can get a sphere of diameter 2r, by placing a fraction of the sphere; each fraction is of infinity size, in each layer. I we use eights spherical arc of radius infinity, one on each layer.
I would then take the layers, rotate each, and then blend all the layers, until what appears is the double infinity, contained in all the layers, but which looks like an infinite sphere of radium 1/2 infinity, in our layer. When you blend layers some things can be reinforced, and others can cancel, until we get the final sphere.
Photoshop layers can blend light/energy. Light/energy is the same stuff 99.999% of the observations of the universe are based on. This layer technique should work. If nature had natural blending layers, between others dimensions and our own, the light and energy signals would not just add or subtract, they can also multiply. What we would see, would not be what is there, in one self absorbed universe. It is a team effort. This is why we need the C ground stat, since this is the same in all universes.
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Drivel.
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lol
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Drivel.
Let me do it this way. Say we have a cube 1M3. How many spheres of radius equal 1 meter can we place in that cube? The answer is less than one. If we took eight 1M3cubes in a 2x2x2 configuration, we could fit one sphere of radius 1 meter in the entire configuration. Each cube would have 1/8 of the sphere. This is where I got eight and 1/8.
Instead of plotting this in one universe; one layer, twice infinity in each of three directions, I stacked them as 8 layers so 1/8 is in of the eight layers. There are many ways to blend these different layers. I did it so the final image appears like the maximum sphere in our visible universe. I saw not point in using an different universe as the standard.
I don't have a licensed copy of Photoshop, or else I would blend the layers, online, to show what happens when light from different sources blend. Since our universe now have another layer, called dark matter and energy, what are the blending consequences? We currently treat light and dark, matter and energy, as separate layers, but in Photoshop, separate layers make it hard to see both layers, since each layer will shield the others, You can see both layers, in partial, when they blend. It comes down to most of the data we collect for the universe is photos of light, which define layers at different distances.
My approach is science art. I present it to make people think.
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The volume of a cube of 1 m radius is 4π/3 ≈ 4.2 m3. So what?
A sphere being infinitely symmetric, if you divide it into n portions by any symmetric means you wil end up with n pieces of v/n volume. So what?
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@puppypower In the past you have made the claim that c (a speed) is the 'ground state' of the universe. Try starting your education by learning what ground state means.
https://en.m.wikipedia.org/wiki/Ground_state
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There would not be enough room in an infinite universe, for a sphere of infinite radius.
Not a problem; the surface of an infinite universe is an infinitely large sphere.
(Unfortunately, that also applies to an infinitely large cube, or (I think) an infinitely large Mickey mouse balloon shaped universe.)
Is there a situation (apart from this trehad) where the possibility or impossibility of an infinitely large sphere actually matters?
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My approach is science art. I present it to make people think.
No, at best it is waffle and it makes people think you have no idea what you are talking about.
In the 3 main sections of the forum we stick with current science, if you want to explore new theories or art do so in the appropriate sections.
Think carefully before your next post or you will be confined to the lighter and general discussion sections.