In 3D, we have a sphere
In 2D, we have a circle
In 1D, we have ?
I thought of this the other day, and the answer is “2 points”
That is, on a line, find the two points that are equidistant from the center point, and those two points constitute your 1D “circle”
This is rather mundane and pointless, but I thought I’d share.
I think it’s interesting since both the 3D and 2D versions look “circular” in some sense, but the 1D version looks nothing like its higher-dimensional equivalents.
In one dimension, can you have two points?
I think I follow your concept.
If you imagine a movie that shows the process of passing the higher dimension object through the lower dimension, then the sphere passing through a plane would begin with a point followed by a series of ever-enlarging circles until the sphere’s diameter is reached, then a decreasing series of circles until that final point is reached.
Taking the circle through the 1-dimension space would involved turning the circle’s plane perpendicular to the line and beginning the pass-through with one point followed by ever-increasing-in-distance-apart pairs of points on the circumference until the circle’s diameter is reached, then an ever-decreasing pair of points until the final point is reached.
Fun video to make sometime. 
A line (which is 1D) has an infinite number of points on it.
I beg to differ. The one-dimensional circle looks perfectly circular in a one-dimensional sense.
Duh… sorry, I was having a brain cramp and forgetting that a circle has one more dimension than a line…
… yes, it’s been a long time since I’ve done any math, why do you ask?
In that case, you’re totally right. I mean, isn’t the derivative of the equation for a spehere a circle? And the derivative of that a line?
I don’t think this is about derivatives. Here are the equations for circles and spheres centered at the origin:
sphere: x^2 + y^2 + z^2 = R^2
circle: x^2 + y^2 = R^2
1D circle: x^2 = R^2
In the last case, the solution is x = +/- R, i.e. two points.
And if you think about it, if you looked at a circle from the side, it would appear to be a line.
Just for fun, here’s MathWorld’s entry on hyperspheres.
Looking at it another way:
In 1D, we have a line; rotating it in 2d space around its midpoint describes a disc
In 2D, we have a disc; rotating it in 3d space on its diameter describes a solid sphere.
or:
In 1D, we have a line; transforming it in 2d space, perpendicularly, describes a flat square
In 2D, we have a flat square; extruding it in 3d space perpendicularly to its surface describes a solid cube.
The way in which objects in n-dimensional space are related to objects in N+1-dimensional space is as much a function of the way they are translated, as it is a function of the shape of the objects themselves. Obvious, I suppose
Hyper-spheres, spheres, circles and such are characterised as the set of points that are equidistant from a reference point.
In 1-D, that would be two points, and the reference point is the midpoint between them, and the “radius” is the length of the chord linking each point to its’ midpoint.
<--------PL---------C---------PR-------->
The “circle” is defined as points(scalars PL and PR), with “center” C and “radius” r=|C-PL|=|PR-C|.
To visualise this, shoot a line through the center of a 3-D sphere. The intersection points of the line with the 3-D sphere induce a 1-D sphere, just as shooting a 2-D plane through the center of a 3-D sphere induces a circle (AKA 2-D sphere).
Probably obvious, but you can do it backwards. Start with a sphere, take a planar section through it anywhere, and the spherical intersection with the plane is a circle.
Take a linear section through the circle (or indeed through the sphere) and the intersection will be two points. Well, unless they’re tangent sections, in which case the intersection (in either case) is only a single point… those correspond to the radius=0 case.
Right, but isn’t the relationship between those equations best shown through integrals/derivatives? That’s all I’m saying (which may well be way off base… as I said, it’s been a long while since I’ve had to do math).
Not really. In general, the equation for the surface (n - 1)-volume of an n-sphere is the derivative of the equation for the n-volume bounded by it, but that’s where the calculus ends.
Ok, volume to surface. That makes sense.
creeps back to humanities-related threads
Didn’t anyone else ever have to read Flatlands in geometry? No? Lucky bastards…
The protagonist was a square, and he lived in a 2D world. A sphere, a visitor from the 3D world, came to visit him, but appeared as a circle, since that was all of the sphere that could be shown without that third dimension. Together, they traveled to a 1D world, which was made up of an endless number of lines, all connected together. They also traveled to a 0D world, which consisted of 1 single point.
My geometry teacher seemed to think this all made sense, and so, I present to you, a listing of the #Ds and their corresponding shape types:
3D: all the objects that we’ve always been told are 3D- spheres, pyramids, cubes, etc. They can move up, down, and back and forth in their 3D world.
2D: squares, triangles, circles, etc. They can move back and forth in their 2D world, as no up and down exist.
1D: lines. All of the lines must also line up into a line. They can move along this line, but any other directions, as well as up and down, don’t exist.
0D: one single point. He cannot move.
If you crucify Jesus on a hypersphere, does he roll off the other side?
A hypercube seems to work okay.
Not necessarily. See Heinlein’s And He Built a Crooked House.
For a deeper look into Flatland, try here. (Note that the title has no ‘s’ on the end.) You can get a plain-text version at gutenberg.org, along with many other classics.
Coming back to this for a second, the reason that circles, spheres and so on look circular is because they have constant (extrinsic) curvature. There’s no corresponding concept in one dimension, and that’s the reason for the discrepancy.