I Set of possible rotations of a 3D object and the rotation history

[Spinnor]

I am trying to understand the picture below which is of a contractible and uncontractible loop in what I would call (proper name?) "rotation space", where "rotation space" is a solid ball of radius π with opposite points on the surface of the ball identified, each point of the ball representing a direction and angle less than or equal to π which could represent the rotation of an object. I understand what a point represents in such a space but I am having a hard time understanding what a path in such a space represents, for example the two paths below. I am thinking that the path represents some kind of "rotation history" of an object but something does not seem right about that idea. I know the answer must be simple but I am stuck.

The above was found at, http://www.damtp.cam.ac.uk/user/examples/D18S.pdf

(which Google Chrome tells me is not secure)

I think my second question is related so I ask it here. I would like to represent the rotational history of an object, say a soccer ball. Let the soccer ball have a spherical coordinate system printed on it. Say the ball is rotating relative to say a soccer field, we will(?) need 6 (edit, 7) coordinates to describe the rotational history of the ball, 2 spherical coordinates give the orientation of the ball with respect the the soccer field, (edit, another angle is needed to to fix the orientation), 2 more coordinates give the orientation of the instantaneous angular momentum vector L of the soccer ball, another coordinate gives the magnitude of L and one more coordinate for time. Is 6 (edit, 7) the minimum number of coordinates to describe the rotational history of the ball or are some of my coordinates redundant? Is such a space the same as S^2xS^2xR^2 (edit, S^2xS^3xR^2) ?

Edit, I should not post questions at the end of the day. If we know the orientation of the soccer ball as a function of time, 3 coordinates plus time, we know the rotational history of the soccer ball, right?

Edit, sorry for mistakes.
Thanks for any help.

 

[haruspex]

I understand what a point represents in such a space but I am having a hard time understanding what a path in such a space represents

Yes, it's not clear.
I know nothing about this area, just replying because nobody else has.

If we pick a point in the ball it represents a rotation about such and such an axis through some angle. If move out along a radius of the ball that gets to a greaterrotation about the same axis.
But what if we move the point a little tangentially within the ball? That becomes an equal rotation but about a slightly different axis. This is like a precession, no? The path is then something like integration of spinors??

Judging from your nom-de-guerre, you know all about those already.

 

[Spinnor]

But what if we move the point a little tangentially within the ball? That becomes an equal rotation but about a slightly different axis. This is like a precession, no?

This is where my blockage is. If the angle does not change but the orientation of the rotation axis changes then an object can not have moved but that can still represent a path in our rotation space. An object that is precessing is constantly rotating so both the angle and orientation change at a constant rate? In our rotation space that path would be a double "conical spiral"? Still not where I want to be. Thanks.

 

[lavinia]

This is where my blockage is. If the angle does not change but the orientation of the rotation axis changes then an object can not have moved but that can still represent a path in our rotation space.

If the angle is zero then changing rotation axis gives a constant path $0⋅N(t)$ that stays put at the origin.

It might be helpful to formalize this a little bit:

Consider the space $S^2×[0,π]$ consisting of sphere of radius 1 Cartesian product the closed interval $[0,π]$. Any path in this space is of the form $(N(t),θ(t))$.

If one identifies all pairs $(N,0)$ to a point then the quotient space is homeomorphic to the 3 ball of radius $π$. The identification map is $(N,θ)→θ⋅N$.

A path of the form $(N(t),0)$ corresponds to changing the axis of rotation while keeping the sphere stationary. It becomes the path $0⋅N(t)$ in the 3 ball. This path is constant at the origin.

Aside: The identification mapping $S^2×[0,π]→B^3$ shows that the 3 ball is the topological cone on the 2 sphere. The cone on any topological space is the quotient space of its Cartesian product with an interval obtained by identifying one of the boundaries to a point. In this case the boundary sphere $S^2×{0}$ is identified to a point. This terminology generalizes the idea of the standard cone which is a cylinder with one of its boundary circles identified to a point. To see this, flatten the cone out by projecting it vertically onto a plane on which it is standing. The image is a circle with a spray of perpendicular radial lines that converge at its center. This is a disk. In general the $n+1$ ball is the cone on the $n$ sphere.

 

[lavinia]

Here is an example of a path of rotations through an angle of $π/2$ with rotation axis the line through $(cos(α),sin(α),0)$. Letting $α$ start at $0$ and end with $π/2$ the path starts with a rotation of $π/2$ around the $x$-axis and ends with a rotation of $π/2$ around the $y$-axis . This path traces a circular arc in the 3 ball.

$\begin{pmatrix} cos^2(α)&cos(α)sin(α)&sin(α)\\cos(α)sin(α)&sin^2(α)&-cos(α)\\-sin(α)&cos(α)&0\end{pmatrix}$

 

[haruspex]

If the angle does not change but the orientation of the rotation axis changes then an object can not have moved

By angle there I assume you mean the magnitude of the rotation of the object.
But I do not understand why you say the object "can not have moved".
Say we rotate the object through angle θ about some axis α. This is represented by a point in the ball distance θ from the origin and with direction corresponding to α.
If we move the representative point so that it keeps radius θ but is now in direction α', the corresponding motion of the object can be constructed as:

1. Rotate through -θ about axis α
2. Rotate through +θ about axis α'

Clearly this produces a net rotation of the object, and the smaller the difference between α and α' the smaller the net rotation.
For algebraic details see above post by @lavinia (but I think there is a sign error in one of the cos(α)sin(α) terms).
To follow the continuous path, apply the small angle approximation.

 

[Spinnor]

But I do not understand why you say the object "can not have moved".

I was confused and didn't understand the little I thought I did (I think you are right, I'm wrong). The simplest of paths I think I understand, for example a line from the origin in the +z direction of length π/2 just represents the continuous rotation from 0 to π/2 about the z axis. It is the curved paths that are more complicated but now maybe I have it. So consider the path below in our rotation space, start at 0 then go to 1,2,3,4 and back to 1. That is, beginning at the point 1 we go in a circular orbit about the origin of radius π/2. So at this point it helps me to pick up a squarish empty plastic container of nuts and perform the rotations of my container given by points 1,2,3, and 4 and also points in between. So as I go around this rotation path my plastic container will smoothly rotate between the orientations 1,2,3,and 4 given below and when it gets back to point 1 it will have the same orientation as it had the first time it got to point 1?

Did I get the path in rotation space of a rotating top that processes in the Earth's gravitational field right? Both the angle and orientation change at constant rates giving some kind of conical spiral path in rotation space?

Thanks.

 

[haruspex]

Did I get the path in rotation space of a rotating top that processes in the Earth's gravitational field right?

Let's say rotations are positive clockwise, viewed from the ball's centre.
If in the cube diagrams the x-axis is positive into the page and the y-axis positive right, then yes.

 

[lavinia]

For algebraic details see above post by @lavinia (but I think there is a sign error in one of the cos(α)sin(α) terms).

There is no sign error. One can check that the matrix has determinant 1 and that its transpose is its inverse.

A good visualization would be to draw the three vector orthogonal frame determined by the column vectors of the matrix and watch how it moves as the angle moves from $0$ to $π/2$.

 

[haruspex]

There is no sign error. One can check that the matrix has determinant 1 and that its transpose is its inverse.

Thanks for checkng.

 

[Spinnor]

This path traces a circular arc in the 3 ball.

Is the following the path?

Thanks.

 

[bolbteppa]

The diagram is just crudely representing a 'continuous' sequence of rotations in 3-D space as a continuous path in an abstract parameter space, you can try to set up a better picture with the line through a sphere (anti-podal points) idea if necessary. The point of the diagram is really to show that paths through SO(3) can be connected but not simply connected. Your counting of degrees of freedom is off, seems like you are conflating the counting of DOF of rigid bodies with the number of parameters needed for a rotation in 3D (i.e. the dimension of SO(3)).

 

[lavinia]

Is the following the path?

View attachment 238708

Thanks.

On the 3 ball digram the curve lies on the sphere of radius $π/2$. Such a curve has the property that it is a path of rotations which all are rotations of an angle of $π/2$ from the initial position. Instead of rotating with a fixed axis, this path changes the axis keeping the angle of rotation fixed.

 

[Spinnor]

Instead of rotating with a fixed axis, this path changes the axis keeping the angle of rotation fixed.

Thank you. This then,

 

[Spinnor]

Your counting of degrees of freedom is off, seems like you are conflating the counting of DOF of rigid bodies with the number of parameters needed for a rotation in 3D (i.e. the dimension of SO(3)).

I was off but now I think I get what the path represents and the important property that a 4π rotation is equivalent to no rotation.

As for mapping the rotational history of an object it seems there are two equivalent ways to do this, a path in "rotation space" or a path in "orientation space"?

Thanks.

 

[lavinia]

Thank you. This then,

View attachment 238730

Right. Although now that I think abut it I did not show that the curve is actually an arc of a circle but only that is lies completely on a sphere of radius $π/2$ in the 3 ball.

 

[lavinia]

@Spinnor Do you see why the loop in the diagram is not contractible?

 

[Spinnor]

@Spinnor Do you see why the loop in the diagram is not contractible?

I thought I knew the answer to that and from the paper it says, "It is clear intuitively that the loop in figure 1b cannot be shrunk to a point while keeping its ends fixed—it is non-contractible." but I am not so sure now. When the paper says "ends" of the loop in figure 1b, are those the points at the origin?

Edit, rereading paper, "ends" of the loop are the points at the origin.

Thank you.

 

[Spinnor]

Do you see why the loop in the diagram is not contractible?

I think I see it now, what would be a good test question to determine if I do "see it"?

Thanks.

 

[lavinia]

I thought I knew the answer to that and from the paper it says, "It is clear intuitively that the loop in figure 1b cannot be shrunk to a point while keeping its ends fixed—it is non-contractible." but I am not so sure now. When the paper says "ends" of the loop in figure 1b, are those the points at the origin?

Edit, rereading paper, "ends" of the loop are the points at the origin.

Thank you.

The intuition is that any contraction to the origin would have to pull the two points on the boundary - at the angle of $π$ - off of the boundary and bring them into the interior of the ball. But since these two points are the same - they represent the same rotation - this would mean that the curve breaks and is no longer a loop.

This same intuitive reasoning can be used to show that if you go around the curve twice the it can be contracted.

BTW: Your book also assumes that the idea of a continuous path of rotations actually makes sense. It assumes that the 3 ball with antipodal points on its boundary sphere identified is topologically the same as the rotation group and is not just a parameter space. You might like to prove this for yourself.

 

[Spinnor]

the rotation group and is not just a parameter space.

I have a feel for the rotation group. Would an example of a parameter space be the space of orientations of a physical object in 3D space? It seems it uses the same space, 3-ball with opposite points on the surface identified, but I don't think it forms a group, the product of two orientations doesn't even make sense?

Thank you.

 

[lavinia]

I have a feel for the rotation group. Would an example of a parameter space be the space of orientations of a physical object in 3D space? It seems it uses the same space, 3-ball with opposite points on the surface identified, but I don't think it forms a group, the product of two orientations doesn't even make sense?

Thank you.

I am. not sure what you mean by an orientation but if you mean an ordered choice of three mutually othogonal directions then these may be interpreted as either a rotation from an initial position or a rotation together with a reflection.

For instance the column vectors in the matrices in post #5 determine an orientation for each angle $α$.

 

[Spinnor]

I am. not sure what you mean by an orientation but if you mean an ordered choice of three mutually othogonal directions then these may be interpreted as either a rotation from an initial position or a rotation together with a reflection.

For instance the column vectors in the matrices in post #5 determine an orientation for each angle $α$.

I think I was confusing two things that are the same thing, thank you.

 

[lavinia]

I think I was confusing two things that are the same thing, thank you.

BTW: There may be more than one terminology but I am used to thinking of an orientation as having one of two values - positive or negative. Each is represented by an equivalence class of ordered bases. Two ordered bases are equivalent if they can be mapped into each other by a linear transformation of positive determinant.

 

[bolbteppa]

I was off but now I think I get what the path represents and the important property that a 4π rotation is equivalent to no rotation.

As for mapping the rotational history of an object it seems there are two equivalent ways to do this, a path in "rotation space" or a path in "orientation space"?

Thanks.

Consider the simpler situation of rotations in the plane where the rotation group is (part of) $SO(2)$ which has $2(2-1)/2 = 1$ degrees of freedom, i.e. one parameter (while $SO(3)$ has $3(3-1)/2 = 3$ parameters). The parameter space is the line $[0,2 \pi]$. A continuous path in parameter space is just a parameter $\theta$ moving along the interval $[0,2\pi]$, e.g. if you rotate around a circle, the path in parameter space is just the angle $\theta$ moving from $0$ to a bunch of other angles...

Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.

 

[lavinia]

Consider the simpler situation of rotations in the plane where the rotation group is (part of) $SO(2)$ which has $2(2-1)/2 = 1$ degrees of freedom, i.e. one parameter (while $SO(3)$ has $3(3-1)/2 = 3$ parameters). The parameter space is the line $[0,2 \pi]$. A continuous path in parameter space is just a parameter $\theta$ moving along the interval $[0,2\pi]$, e.g. if you rotate around a circle, the path in parameter space is just the angle $\theta$ moving from $0$ to a bunch of other angles...

Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.

To continue this example, the loop that connects $0$ to $2π$ is a circle and cannot be shrunk to a point without breaking the circle,

Unlike in the case of $SO(3)$ no multiple of this loop is contractible. In $SO(3)$ the double of any loop is contractible ("null homotopic").

 

[Spinnor]

Regarding "rotation space" and "orientation space", I am not sure these things make sense and again it seems like you are thinking of rigid bodies where one fixes the position of the center of mass then the orientation of the rigid body about that center of mass.

I was trying I was using the wrong terminology and was confused but a little so less now, I think, and I was thinking of a physical object with a fixed point.

Say we have two right-handed rectangular coordinate systems that coincide. Now say we rotate one of the coordinate systems in some complicated manor keeping the other coordinate system fixed. The rotated coordinate system has a new orientation, relative to the the fixed coordinate system, which can be represented as a point, p, of our 3-ball and the rotation "history" of the rotated coordinate system is represented as a path in our 3-ball starting at the origin and ending at the point p. What I called the rotation history is just the orientation history. From the Wiki article on orientation it looks like there is more than one way to describe the orientation, such as the matrix of unit vectors that Lavinia listed above.

I am in a better place now in regards to rotations. Thanks.

 

[lavinia]

The diagram is just crudely representing a 'continuous' sequence of rotations in 3-D space as a continuous path in an abstract parameter space, you can try to set up a better picture with the line through a sphere (anti-podal points) idea if necessary.

From the topological point of view the representation is precise. The map from the 3 ball with antipodal points on the boundary sphere identified into the rotation group is a homeomorphism.

The point of the diagram is really to show that paths through SO(3) can be connected but not simply connected.

And also that the double of any loop is null homotopic. So the diagram implies that the fundamental group of $SO(3)$ is $Z_2$.

 

[bolbteppa]

I was trying I was using the wrong terminology and was confused but a little so less now, I think, and I was thinking of a physical object with a fixed point.

Say we have two right-handed rectangular coordinate systems that coincide. Now say we rotate one of the coordinate systems in some complicated manor keeping the other coordinate system fixed. The rotated coordinate system has a new orientation, relative to the the fixed coordinate system, which can be represented as a point, p, of our 3-ball and the rotation "history" of the rotated coordinate system is represented as a path in our 3-ball starting at the origin and ending at the point p. What I called the rotation history is just the orientation history. From the Wiki article on orientation it looks like there is more than one way to describe the orientation, such as the matrix of unit vectors that Lavinia listed above.

I am in a better place now in regards to rotations. Thanks.

Maybe this will help - one can take the perspective that rotations are "active" or "passive", one view is that they are rotating vectors in a fixed coordinate system, the other view is rotating the coordinate system while leaving the vector fixed.

https://en.wikipedia.org/wiki/Active_and_passive_transformation

The active 'rotation history' of rotating some vector is basically the path the tip of the vector threads out as it's rotated, or rather the inverse image of this (i.e. in 'parameter space'), while the passive 'rotation history' is the same path in parameter space, but this time the image of the path in parameter space is not a vector threading out a path, it's the coordinate system rotating around.

 

[Spinnor]

The active 'rotation history' of rotating some vector is basically the path the tip of the vector

But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.

 

[lavinia]

But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.

Right. The motion of a single vector does not determine a rotation. It merely describes a path on a sphere centered at the origin.
A rotation is described by a rigid motion of an orthonormal frame.

A rotation of coordinates is a rotation of basis frames of the coordinate system.

I do not see what this has to do with spinors, Can you explain?

 

[Spinnor]

I do not see what this has to do with spinors, Can you explain?

I think I read that spinors don't live in our 3 dimensional space but they can be represented in our 3 dimensional space by a flagpole and flag (vector and flag) plus a sign,

but the flagpole and flag is mathematically the same as a rectangular coordinate system, we need a direction and an angle to determine their orientations relative to some fixed coordinate system as above?

Thanks.

 

[bolbteppa]

But the vector does not record rotations about its axis so can we please attach a ridgid flag to the end of the vector? With the flag we can then record and note rotations about the length of the vector. Then it starts to look like a spinor, my favorite unknown.

Thank you.

Unfortunately, spinors are not expressing the idea of a vector rotating about it's own axis, they are related to connectivity of paths in e.g. $SO(3)$ and the necessity of (in this case) dealing with $SU(2)$ rather than $SO(3)$ as the pdf of the OP is trying to explain. What I have read of this flagpole stuff seems to assume what a spinor is and then randomly uses flags to represent it.

 

[Spinnor]

Unfortunately, spinors are not expressing the idea of a vector rotating about it's own axis

The flag is supposed to be rigidly attached to the flagpole. So if we rotate the vector about the vector axis the flag will rotate as well. But the flag and flagpole convey no more information than a right handed coordinate frame so I am not sure why it is used( edit, the spinor "flagpole" can have a length, an extra bit of information that the coordinate frame does not have)?

What I have read of this flagpole stuff seems to assume what a spinor is and then randomly uses flags to represent it.

From the little I understand the flag is a necessary part of the representation. See,

https://arxiv.org/pdf/1312.3824.pdf

Thank you.

 

[lavinia]

The flag is supposed to be rigidly attached to the flagpole. So if we rotate the vector about the vector axis the flag will rotate as well. But the flag and flagpole convey no more information than a right handed coordinate frame so I am not sure why it is used( edit, the spinor "flagpole" can have a length, an extra bit of information that the coordinate frame does not have)?
From the little I understand the flag is a necessary part of the representation. See,

https://arxiv.org/pdf/1312.3824.pdf

View attachment 238886

Thank you.

@bolbteppa is correct. The flag just gives the rotation amount around the axis. This is not a spinor. As the book says there is also a sign. But this method of visualization does not give a clear definition IMO. For every rotation there are 2 spinors that correspond to it. These have opposite sign in the group of unit quaternions.

 

[Spinnor]

The idea is used by several authors, those that are still alive could better defend its use, not I. I think they all mention the need for another bit of information, the sign. Thank you.

 

[bolbteppa]

If the flag is to not be completely superfluous it should at the very least capture that a $2 \pi$ rotation brings in the minus sign, however (page 6):

"The angle doubling leads to the curious feature that when $\theta= 2 \pi$ (a single full rotation) the spin rotation matrices all give $-I$. It is not that the flagpole reverses direction — it does not, and neither does the flag — but rather, the spinor picks up an overall sign that has no ready representation in the flagpole picture."

My reading of MTW (page 1157 on from your last post) is again that they do not give you spinors from this idea of flags, they try to describe the already-established notion of spinors using flags, and earlier (very badly) motivate spinors by arguments that amount to the arguments of the PDF of the OP.

 

[Spinnor]

All right then, into the trashcan of science with the flagpole and flag. I wish there was something just as intuitive to replace it.

Thank you.

 

[bolbteppa]

I would be shocked if one could get more intuitive than the notes of the OP for this stuff

This perspective of spinors from the POV of connectivity of Lie groups is very important for understanding why (finite dimensional) spinors exist for the subgroup SO(n) of GL(n) but not GL(n) itself

https://www.physicsforums.com/threads/why-there-are-no-spinors-for-gl-n.240240/

A nice simple proof of this would be welcome (hint hint readers)

 

[lavinia]

I thought it would be helpful to elaborate on the parameterization of $SO(3)$ given in post #1. This is the 3 dimensional ball $B^3$ with antipodal points on its boundary identified.

Topologically $B^3$ modulo these identifications is homeomorphic to the real projective 3 space $RP^3$. The book is showing that $SO(3)$ viewed as a topological space is homeomorphic to $RP^3$.

In topology, the real $n$ dimensional projective space $RP^{n}$ is the quotient space of the $n$ dimensional unit sphere in $R^{n+1}$ with its antipodal points identified. Antipodal points come in pairs and are the intersections of lines through the origin with the unit $n$ sphere. They may be thought of as opposite geographic poles. The quotient mapping $S^{n}→RP^{n}$ that identifies antipodal points is 2 to 1, is continuous and any small enough ball in $S^{n}$ - a ball that contains no antipodal points e.g. a polar ice cap - is mapped homomorphically into a small ball in $RP^{n}$.This follows from the definition of the quotient topology and shows that $RP^{n}$ is a closed $n$ dimensional manifold and that $S^{n}$ is a 2 fold covering space of it.Consider now the northern hemisphere of $S^{n}$ together with the equator - which is the $n-1$ dimensional sphere $S^{n-1}$. This space is an $n$ dimensional topological ball as can be seen by projecting it vertically onto the $n$ dimensional plane containing the equator. Under the quotient map $S^{n}→RP^{n}$ antipodal points on the equatorial $n-1$ sphere are identified and in the northern hemisphere proper, each point is mapped to a unique point. Every point in $RP^{n}$ is represented either by a unique point if the northern hemisphere or by a pair of antipodal points on the equator. In the case of $S^3$ the northern hemisphere is a three dimensional ball So its image in $RP^3$ is a three dimensional ball with antipodal points on its boundary 2-sphere identified. This is exactly the same topological description and in post #1 and shows why $SO(3)$ is topologically the same as the real projective space $RP^3$. So $SO(3)$ is more than just a group of matrices. It is a closed three dimensional manifold.

The only difference between this description and the description in post #1 is the interpretation of parameters. Instead of axes of rotation, one has directions along great circles emanating from the north pole. Instead of an angle of rotation, one has the distance from the north pole. The angle $θ$ lies between $0$ and $π/2$ rather than between $0$ and $π$. Points along the equator all have an angle of $π/2$. The curve illustrated in post #1, the straight line through the center of the ball with end points on the boundary, corresponds now to a half great circle through the north pole. Its end points are antipodal so it projects to a closed loop $γ$ in projective space.

 

[lavinia]

Closed loops in projective space

As was shown in post #40 projective space $RP^{n}$ is the quotient of the $n$-sphere $S^{n}$ with antipodal points identified. A loop in $RP^{n}$ lifts to two antipodal paths on the sphere and beginning and ending points on one path are antipodal to beginning and end points on the other. If the paths are also closed loops the lift is two antipodal closed loops on the sphere. If the paths are not loops then the end points of each are antipodal. In this second case, the two paths fit together to make a single loop.

So every closed loop in projective space is covered twice by its preimage on the sphere. The preimage is either two antipodal loops or two antipodal paths that join together at their antipodal ends to form a single loop. In this second case, the closed loop does not project back onto the original loop but instead projects to its double, the loop that wraps around it twice. This is because each piece wraps around it once.

Each of these loops is contractible - in fact every closed loop on a sphere of dimension greater than 1 is contractible. - and any contraction projects to a contraction in projective space. A contraction of either of the antipodal closed loops projects back to a contraction of the original loop that they come from, while a contraction of the. spliced together loop projects to a contraction of the double of the original loop.

So every closed loop in a projective space of dimension greater than one is either contractible or its double is contractible. The double of the diameter line in post #1 is contractible but since it is not closed in the 3 ball, it may not be contractible by itself.

The idea of the proof that every closed loop on the sphere is contractible.

If a closed loop on the sphere misses at least one point, then it can be contracted along great circles through one of the missing points. If the loop is space filling, then it can be first continuously deformed into a loop that is not, then contracted. Proving that a space filling loop can be deformed to a non-space filling loop requires a little work and is the only hard part of the proof.

Notes:

- A space filling curve is a continuous path that completely covers a region of space. Every point in the region - for instance of a square or of a cube. - is crossed by the path. Such paths can be shown to exist as the uniform limits of certain sequences of continuous paths.

- The formal definition of a contraction of a loop

One imagines a contraction of a loop as a stretched rubber band that shrinks as it releases tension. At each point in time, the band forms a smaller loop until finally it has zero tension. Formally this is a continuous 1 parameter family of loops and may be described as a continuous map from $H:S^1×[0,1]→X$ from a circle Cartesian product an interval into a topological space $X$ which at time zero is the starting loop and at time one is the constant loop. The map $H$ is called a homotopy and is similar to a variation except that it is only required to be continuous rather than smoothly differentiable and end points are allowed to wander. The idea of a homotopy is not restricted to $S^1$ and makes sense for any topological space.