With a single needle you can represent all the basic dimensions…
Zeroth dimension: point of needle
First dimension: shaft of needle
Second dimension: spin needle along a single axis until it blurs into circular plane
Third dimension: spin needle along horizontal AND vertical axes to get blurry sphere
Time dimension: aging of the needle
I don’t know if this has any use (other than being kind of cool). I’m curious if there are any videos of this for the 2nd and 3rd dimensional representations above. Also, I wonder how much faster the needle must spin to form a sphere vs. a circle (my intuition says twice as fast).
To get a sphere, you’ll have to spin it in different directions at different rates. And how much faster it’d need to be would depend on the dimensions of the needle, but the proportion would be somewhere in the vicinity of (length of needle) / (width of needle).
Of course, you could get as many dimensions as you want by spinning around more axes. How to get a needle to spin along those other axes is an exercise left for the reader.
For a sphere you’d have to spin the needle on one axis at the circumference of the circle divided by the diameter of the dull end times the rate of the the other axis. You have to use the dull end because the point on your needle has 0 dimensions. You can also represent 3 dimensions by any movement on 2 axes.
**Chronos **- how does spinning on more axes get you more than 3 dimensions?
TriPolar- Thank you for pointing out that only the dull end of the needle will be functional in generating the third dimension. I hadn’t even thought of that! I’m still digesting the math portion of your answer.
I’m curious why you think speed has anything to do with this. As long as the needle stays in a plane, it will make a circle. In order to make a sphere, it has to come out of that plane, and meet all points on the sphere’s surface. Eventually, it will happen (depending on how the needle moves), but speed is irrelevant.
Yeah, it would just be the number of rotations of one axis times one rotation of the other to cover all the points. I guess the idea of making a blurry image based on persistence of vision brings the speed into it. That would assume the needle itself is 2 or 3 dimensional.
But if you are just defining the points, all you need is a one dimensional needle, where the 0 dimensional point represents a point in space, and the length of the 1D needle is a radius that defines a circle and a sphere. It also defines all the regular polyhedra with an additional multiplier. Add more data and you can define all the semi-regular shapes, and with a little more data, some fractals.
Two needles give you the all the conic sections, if the length of the needles is infinite. The ellipses can be done with finite length needles.
To clarify a bit for the needle creating a sphere through blurred rotation: I was imagining the axis of rotation as the center point on the shaft of the needle. I don’t know if this is any more efficient than placing the axis on the point of the needle, but my mental image was the former.
The rotational transformation of a needle to trace a sphere over time is relatively pretty easy:
All axes are centered on all dimensions of the needle:
Let the X axis be perpendicular to the shaft of the needle, and rotates on the Y/Z plane. A rotation time of 1 Unit.
Let the Y axis be parallel to the shaft of the needle. It rotates on the X/Z plane at a rotation time of 1.75:1 units.
Let the Z axis be perpendicular to the shaft of the needle, but rotate on the X/Y plane at 3:1 time units.
So, I set this up in my CG software, and let the needle point (orange) and the dull end (cyan) trace its path over time… and… Here’s my result.(Quicktime .mov / 17 MBs / 1.5 mins)
ETA: I’m assuming over enough time (perhaps infinite (!) ) the needle would trace a perfect sphere. I can’t prove this function wouldn’t start repeating its trace – perhaps some math whiz can figure it out?
Every dimension will have another axis associated with it. So if you can manage to spin the needle around a fourth axis, you’ve got your fourth spatial dimension right there.
Of course, in a three dimensional space, there’s no fourth axis to spin the needle around. But the point is, if you could spin it around anothre axis, you would be in a higher-dimensional space.
If the needle is spinning atop a sheet of glass and you move the glass, does that count as a different dimension?
If so, that’s 2 more easy dimensions to add and still able to visualize via mental model. (A. move the glass in a circle along its plane and B. move the glass with a vector component perpendicular to its plane)
Forget spinning entirely and use a nail. For the second dimension, look at the nail head straight-on such that it looks like a circle. For the third, look at it at an angle such that the shaft runs away from you, and you can see depth, length, and height.
OK, this is about the definition of axis then. I consider there to be infinite axes within 3 dimensional space. And the 4th dimension is not based on rotation on an axis at all.
My current thinking is that the ends don’t really matter. If the needle is spinning super fast around the horizontal/vertical axis of the center of the shaft, my hunch is that the shaft itself (allowing it some thickness) can create the blurry sphere, for all intents and purposes. This is because it will look like a circle from any angle. (Moreover, a spinning circle creates a blurry sphere.)
Of course, this would negate the validity of the 1st dimensional quality of the shaft, so so much for Platonicity. We’d be required to make one end possess some thickness for the whole thing to be formally valid (shame though).
Chessic Sense - (I’ve always loved that word “Chessic”) Your nail idea is appealing for its simpler 2d rendering of the circle, yet its 3d rendering seems a little weak. The elimination of motion aspect is interesting and this would be very good for live demonstration. Nevertheless, some beauty seems to be lost here in losing the sphere and the artificial circle…hard to explain.
No prob… I love toying with visual, geometric ideas like this, so long as I don’t have to do any math. I also hobby in creating CG science-based animations (currently working on a cool and dynamic scale of the solar system demonstration)
I was thinking about this as well. Take a one dimensional line (axis in its center), and spin it perpendicular to it’s length. If it has infinite spin, you essentially have a perfect 2-dimensional disc… Now, take this “virtual” 2D disc, and spin that along the axis that is parallel to the original one dimensional line you started with. At infinite speed, you have a “virtual” 3D sphere.
To talk of blurriness, as in the “persistence of vision” sense (or even frame/exposure motion blurring found in photography) would be fine to a degree in the real world, but in the pure geometric sense, I don’t think there’s any accounting for it (?).
Are we making a definite distinction between the 4th dimension as spatial or temporal? I hear a lot of people who conflate the two, but they are very different things. If in the OP, you we’re talking of the “temporal” 4th dimension – that is, time – then you’re already demonstrating it when you introduced spinning the needle to create a 2D disc, rendering the aging of the needle redundant (however, still another way to look at it).
Anything above and beyond 3 spatial dimensions, is sort of over my head (yet, I’m still fascinated and always try to grasp and understand them).
One last thing: Are you perusing some sort of visualization aide in helping others picture what dimensions are?
Yes and no. The common assumption among scientists and mathematicians in these discussions is that all axes are taken to be linearly independent of all other axes. Essentially that means each new axis you add must be perpendicular to all the others. That forces a 1-to-1 mapping between axes and spatial dimensions.
Technically, none of the dimensions require any rotation. It’s just that you’ve chosen rotation of a lower dimensional object to illustrate other spatial dimensions.
Well this is difference between computer graphics and mathematical geometry. We have a granularity in the real world based on pixels and the mechanism of our eyes, so nothing is really 0 or 1 dimensional in representation, and in reality everything is 3 dimensional (except for black holes maybe). Innumerable CPU cycles have been wasted computing graphics at greater resolution than necessary since the minimum is really one pixel. Luckily CPU cycles are a growing and limitless resource. But there have been problems in some information with a ‘disappearing star’ effect. Accurately modeling stars many light years away gives them an actual size much smaller than any pixel and an algorithm that does not account for this may have them blink in and out of virtual existence through spacial transformation.
Anyway, cool video. The random-ish path of the needle point will tend to make a blurry image that is more spherical in appearance at a slower rate of rotation than the simplest progression along two axes of rotation, and distribute the overlap the path of the path of the needle end around the sphere instead of having it concentrate at a pair of poles.
cmyk, in order for your needle to trace out the full sphere (technically, never, but I’m parsing that as “for any given point on the sphere and any given nonzero distance from that point, to eventually get within that distance”), the ratios of your rotation speeds need to be irrational. What you’ve got there will repeat itself after 21 of the shortest rotation, if those are the actual numbers you used.
I have no idea, but that’s interesting (paramount to me are concrete visualization aides for my own mental images). Basically, someone asked what a “dimension” was, and this popped in my head as a cool little thought experiment. I like the line of thought that simply adding an axis of rotation creates an added dimension, although this only seems to apply when moving from 1d to 2d and 2d to 3d.
I also hobby in creating CG science-based animations (currently working on a cool and dynamic scale of the solar system demonstration)