A while ago whilst doing the washing up I started to think about the relationship of circles and spheres to squares and boxes. This led me to come up with a little recipe for concocting a different way of looking at the relationships. It goes as follows:-Recipe for Occulted Pie

Pi, as we all know, is the manifestation of the relationship between the diameter and the circumference of a circle. This relationship brings about the ability to calculate the area of a circle, the volume of a sphere and the surface area of a sphere or, by division, parts thereof.

Pi is assigned the value 3.141592654 etc. but for the sake of convenience can be stated as being 3.1416 bringing it to four places of decimals. If we start from the point of calculating the area of a circle we would use the formula pi x r² . But, if we take the said circle and place it in a square so that the circle touches each side of the square without cutting it, we can calculate the area of both square and circle and then compare one area with the other to find the circle to square ratio. This happens to be 0.7854:1. The area of the circle can now be calculated by using the formula 0.7854 x d² , where d is the diameter of the circle. This proportion is one quarter of Pi or pi/4. We now have the first occulted pie.

From here we can move on to calculating the volume of a sphere. By using the same method as before but instead of a circle in a square we can now place a sphere in a box and work it out from there. The volume of a sphere is calculated by using the formula (4/3)pi x r³ and the volume of the box is length times height times depth. By comparing the results we come up with a ratio of 0.5236:1 which means that we can now use the formula for finding the volume of a sphere as d³ x 0.5236. This proportion just happens to be one-sixth of pi or pi/6. This is the second occulted pie.

We now turn to the surface area of a sphere. The surface area of a sphere is given by the formula 4pi x r² and the surface area of the box is arrived at by squaring one side and multiplying by 6. When this is done the proportionality is found to be 0.5236:1. Therefore it follows that the surface area of a sphere can be found by using 6d² x 0.5236.

Does anyone else have a recipe for some other kind of numerical dish that I can add to my cook book?

Arthur Pickford

Try the golden ratio. Its far more interesting. I think its 1.618 to some places.
Should be (1+sqrt5)/2

Try solving n² - n - 1

Just a generalisation on Arthur's results of the relationship between the volume of a circle/sphere and a box, for higher dimensions:

In an n-D space the hyper-sphere is defined as the locus of all points with the same Euclidean distance from the centre. A box has volume (2r)ⁿ .

I know we're all supposed to be very sophisticated and everything, but what a load of rubbish N-d space is. It is fine in my opinion to have n variables in equations and things, but you can't very well draw a graph of them can you is n is bigger than 3. You have to draw a graph of three of them when one is fixed.

I'm sorry if I've ignored anything (I have no knowledge (or interest) in N> 3 dimensional space) but it really is no good.
Please RSVP and tell me how wrong I am

Neil M

Neil...

General relativity: 4-D spacetime.
String Theory: 11-D or 26-D spacetime.
Quantum Mechanics: Infinite dimensional vector spaces.

The study of topology (i.e. shapes) in high dimensional spaces is becoming increasingly important in many areas of theoretical physics...

Being able to generalise results we can easily visualise to spaces we can't is perhaps the single most important thing maths does...

Sean

If you noticed, I was careful to say "what aload of rubbish N-d space is"
I am well aware of the 4d spacetime. However, this is slightly different as the fourth dimension of time is a little separate from the other three. Taking dimensions as just meaning variables, topology and equations p + q + r + s = d etc is fine, but calling it space, and saying hyperspheres is a bit off.

Regards
Neil

Neil - I completely disagree with the point you are making.

I think that Sean's last point is extremely important. Never mind about spaces - maths is all about generalisation and new ideas. In my opinion this is just as if not more important than problem solving.

Maths takes ideas we are familiar with intuitively and then extends their scope into the non-intuitive, while retaining their defining properties - sometimes by making an analogy. Even things that you may think are totally abstract are initially created out of ideas we are familiar with - they have to be otherwise they'd be impossible to define. Everything we can imagine is based on analogy with the obvious.

Now it is completely vital to be flexible with how you handle these new systems. As long as their definition doesn't involve writing down a contradiction (which I assure you N-d space doesn't) rejecting their usage is backward thinking and completely counter-productive. I will be the first to admit that the axioms of maths are heavily shaped by the physical universe. But just because the rules when extended do not appear to directly apply, this does not mean for one minute that the extension is not useful - even for solving problems that can be stated within the original system. There are loads of examples of this, the most obvious being complex numbers which can be used to solve problems that don't involve complex numbers.

I'm sure you've heard the story of Cantor's ex-tutor. When Cantor came up with the idea of infinite sets (whose properties were perfectly well defined, consistent and based on analogy with familiar ideas of numbers and sets) his tutor not only refused to work with the new ideas but also demanded that irrational numbers should be rejected and maths should return to that of Pythagoras' time. And it is extremely easy to fall into this trap. After all, numbers were invented to describe physical quantities and to do this there is absolutely no need for irrational numbers. Any quantity is an approximation and rational numbers can make as good an approximation as you should choose. But without irrational numbers, we would not have got very far in analysis and pure maths would have suffered. And they can be used to solve problems which can be stated within a system in which they do not exist.

Now onto the subject of hyper-space. A dimension simply means the degree of freedom of a variable. It was invented to describe what we observe in the universe - the position of an object. The reason it works nicely is the independence of the dimensions. You canot change your height by moving forward, backward, left or right.

Therefore we can represent the position by three separate varibables or dimensions x y and z. The n-D space means that n of the dimensions are allowed to change.

Imagine we have a 1-D space (a line). If we keep the dimension constant we get a 0-D space (a point). For a 2-D space, if we keep a dimension constant we get a 1-D space. The same for 3-D and 2-D. Continuing inductively, we see that a n-D space with one dimension kept constant is a (n-1)-D space. There is absolutely no reason to stop at N = 3 - to do so is letting the restrictions of the physical universe spoil a pure mathematical idea that was only originally designed to apply directly to the physical universe. I'm not sure what you mean when you say that "you can't very well draw a graph of them if N> 3": firstly you can represent 4-D in a 3-D drawing in much the same way that a 3-D picture can be represented on a 2-D paper. Secondly - it really doesn't matter what you can draw a graph of: a graph is a construction within the laws of physics. If an idea doesn't fall within this realm this does not mean that it is wrong. It means that maths is a whole lot more general than physics. (Well classical physics anyway.)

Now you may be wondering how on Earth the idea of volume can generalise to "hyper-volume". Well length, area and volume are designed to describe the amount of space contained. The hyper volume in the n-dimensions is the summation of the n-1th dimension as the nth dimension is changed. This makes sense because the amount of space must be additive. So my entry in the above table at N = 4 refers to 4-D hyper-sphere (that is the locus of points whose Euclidean distance is constant; Euclidean distance incidentally is defined using the idea that an (N-1)-d space is an N-d space with the Nth dimension constant). It says that if you randomly pick a set of co-ordinates from within the N-d hyper-box then the probability that the point will lie within the hyper-sphere is pi^2 /32. That was what I wrote in to say.

Yours,

Michael

Just to add a little more to what Michael has said. Besides its mathematical intuitiveness, N-d space, even in the simplest possible generalisation from our 3-d environment, is used all over the place in mathematics and physics and is very useful. Many problems that appear as though they merely involve many variables can be given an intuitive and conceptually useful geometric setting in N-d space. From linear problems involving n by n matrices to string theory, in which 25 of the 26 dimensions are spacelike (i.e. correspond to what you are calling "space"), the geometric idea of generalising our immediate world is essential.

Sean

Once again, I originally said "what a load of rubbish N-d space is ..."
Taking dimensions as just meaning variables, topology and equations p + q + r + s = d
etc is fine"

To state clearly, more than 3 variable is fine. Calling it space is just wrong.

Michael said: "A dimension simply means the degree of freedom of a variable" which is like what I said before. This definiton can help with many things. The object we call space has 3 such dimensions in itself, and an associate of time. The rate of a chemical reaction may depend on the concentration of n reactants (with relevant indices). However, comparing the use of dimensions in general to space by calling things hyperspheres and 'volumes' is counter-productive; it makes it less acceptable because we are used only to 3d space. I think this is what you thought was bugging me initially, and I apologise if my message gave that (incorrect) impression.

Regards

Neil M

I still don't think there is a useful distinction here. Space... variables - they have the same properties and by looking at the system both ways you can gain extra insight. Remember unless writing down the definition of a system involves a contradiction there is absolutely nothing wrong with it. (And that incorporates hyperspace and hyperspheres.) What exactly is bothering you?

Michael

Please define 'space' and say exactly what it has apart from 3 degrees of freedom, a topology and an inner product.

In analysis, a vital concept is that of the n-ball, i.e. a hypersphere (with the inside), as it gives an idea of how close points in the space are (i.e. the n-ball of radius epsilon at a point is all the points a distance less than epsilon from the original point). The idea is used everywhere : from covering a space with n-balls of a given radius to showing that a sequence converges by showing that the points in the sequence get arbitrarily close to some given point. It is a fundamental concept in proving things. It is so useful because it is easy to visualise , because it is a natural extension of 3 dimensional balls.

Sean

This is really getting a bit silly. Neil, what characterises space to you? When mathematicians talk about space they don't mean something physical, they mean idealised (abstract) objects (or ideas) with certain characteristics. For instance, a vector space is a set of points which can be added together to form new points. As far as we can tell, the universe has some characteristics of a vector space. A normed space is one in which the points all have a length. A metric space is one in which the points each have a distance between them and other points. A topological space is one in which you can't even decide on distances between points. How do you know which of these best characterises our universe? Maybe none of them do? Why do you think the universe is 3 dimensional? Do you think 2 dimensional spaces are ok? Of course, there is no 2 dimensional object in our universe, does that make 2d space useless? Dealing with ideas that are counter to our intuitive ideas of what the world is like is what has enabled modern society to be like it is. When Newton came up with his theories, many people at the time hated them because they didn't like the idea of "force at a distance", how does that fit in with your idea of space? Einstein's relativity has shown that the Euclidean idea of space doesn't apply to the real world, except approximately. In quantum physics, you do away with the idea of particles entirely, instead there is one all encompassing wave function ! Sorry if that sounded a bit aggressive, but I wanted to make the point that intuition sucks!

A slight qualification...I wouldn't say intuition sucks. It's just a question of distilling what is general in intuition to what is contingent to our immediate environment. Clearly the intuitive idea of distance is useful in more general settings, although equally clearly there are spaces in which it is best not to think in terms of distance.

Michael- Before we acquire a mathematical understanding of space, we of course experience it ourselves, and become familiar with its structure, and how many dimensions it has that we can vary ourselves. Then, after we have a mathematical understanding of 1, 2 and 3 d space, (which for various reasons, is very easy to visualise and compare to our natural world) we look at nd space. This can't very well be similarised in the same way to our natural world. Yet the language and meaning is identical. This is what makes a daunting subject initially for many (myself included). After a little help and reassurance, we can get to grips with it a little better (I haven't done this yet). What is bothering me is that this step is unneccessary. Because our natural world obeys the same rules as n-d space, there is a perceived neccessity of comparison between the two.

Sean - my initial (non-mathematical) definiton of space is: what an object occupies in order for it to exist. Dan has made the distinction to what I have been calling space, and what you have been calling space. Obviously, they are the same, but since I have no knowledge of nd techniques for n> 3, I have not been able to visualise the identicity [is that a word? You know what I mean:)]. I still believe that it is a bit inconsiderate to call our natural space and mathematical space the same thing, even if they are the same thing! It doesn't help the understanding.

Dan- surely it is the wave-particle duality that makes quantum physics different! Everything in our natural world (including quantum stuff) is all based on chance. The high-level equations are all probablilities. This upset Einstein, who famously said: "God does not play dice". And intuition does not suck! How do you think we both got our idea of what space was in the first place?

I am having trouble visualising a (say) 4d sphere. Please help me to do so. What does it look like? Perhaps it goes beyond looking like anything.

2D: ax + by = c is a line
3D: ax + by +cz = d is a plane
4D: ap + bq +cr +ds = e is a what?

RSVP

Regards

Neil M

Just thought: Is it a cube?

First. ax+by=c is a line in 2D, but it is a plane in 3D (you need two equations to specify a line in 3D). And it is neither a line nor a plane in 4D etc. The final equation is easiest to discuss if e=0 (this isn't much of a restriction, it just requires a translation). It certainly cannot describe a cube because a cube is not a linear subspace of R⁴ . It will descibe a 3D linear subspace of R⁴ (this means that if the points (p1,q1,r1,s1) and (p2,q2,r2,s2) satisfy the equation then so does their sum). In fact, the equation describes a space (i.e. a copy of R³ )!

Second. You say: "space is what an object occupies in order to exist". Well, for a start according to string theory this means space has probably 11 dimensions.

Third. What is so magically different about 3 dimensions? This is a valid physical question but if you start from the premiss that 3D is natural and everything else artificial then you aren't in a position to ask the question properly.

Just to clarify, when I said "intuition sucks" I didn't mean what I said. In fact I am an intuitionist in mathematical thinking. What I meant was that everyday intuition sucks, the intuition that says space cannot have more than 3 dimensions, the intuition that makes people play the lottery, the intuition that says the Earth is flat, etc.

Maybe it's one of those things you accept as you get more used to it. When I was 14 or 15 I used to maintain that 0.999... was NOT 1, because the representations are different. I think I also used to think that different levels of infinity was a lie. What an irritating little person I must have been to my teachers.

On the subject of visualising 4-D spaces - try the following. Our basic problem is that because of physical constraints we can only see 3 dimensions. Now imagine the analogous situation where you find yourself in a 3-D space but only have the capacity to see in 2 dimensions. Let's say that you can see the directions left, right, forward and backward but not up or down. Now imagine you are standing next to a 3-D sphere. You can see a circle. (This is because any 2-D slice through a sphere is a circle.) Now pretend you are standing on a lift which will increase your height uniformly. As we get closer and closer to the diameter of the sphere the circle will get larger and larger until eventually it reaches a maximum. Then it will start decreasing again.

Now imagine you can see in 3-D as usual but find yourself in a 4-D space, next to a 4-D sphere. You are on a lift that will change your position in the 4th dimension that you cannot see. What will you see? You will see a 3-D sphere. This will increase in radius as you move up on the lift until you reach the diameter when the radius of the 3-D sphere will start decreasing.

In fact it was basically using this method that I worked out the hyper-volume of the n-D spheres in the above table. All it involves is simple calculus. Hyper-volume, I guess, can be thought of in terms of probability. (If you pick a set of co-ordinates at random then the probability that they turn up within a space of hyper-volume V is proportional to V.)

In my opinion the most important thing to remember is that physical space inspired a pure-mathematical system, which works in the same way as physical space but is a lot more general. The space in our universe is simply a special case where we set N = 3 (with classical physics anyway).

Best wishes,

Michael

Michael-

This way sort of helps to visualise it, but it is difficult to imagine only seeing 2D to start with. I still don't see why it is so easy to visualise.

Everybody-

OK. Suppose we are using 4D space. Usually when we draw graphs of space (2d and 3d), we have the dimensions perpendicular. How can 4D dimensions be perpendicular? This may sound stupid, but remember I'm still a beginner at this! Also, can you model a 4d 'graph' in 3d space in the same way we draw X,Y,Z axes on 2d paper? OK so paper is obviously 3d, but we can't draw on the thin side:).

I suppose our definiton of perpendicular for nd space is that two or more dimensions are perpendicular if you can change one without affecting the others.

Even more bizarre, suppose we didn't use conventional dimensions as straight. Suppose we had a curved (to a certain degree)dimension. Then arcs would be straight-lines in our notation, and vice-versa.

Please tell me if any of the above has any similarities to nd space.

Regards

Neil M

Have any of you read 'Flatland' (I can't remember who by). It helps me when I think about dimensions.
Richard

Does cross-product work as well? (I assume you just showed that the dot product carried)

ie, in 3d space,

Not quite. If 4D space, there is a 2D space of points perpendicular to any 2 nonzero vectors. For instance, u=(1,0,0,0) v=(0,1,0,0) then both of (0,0,1,0) and (0,0,0,1) are perpendicular to u and v. You need 3 vectors to make it work, you could define an "4d cross product" I guess, written [u,v,w] or something like that, but there might not be a nice formula for it like there is for the 3D case. Also, how would you specify the length? In 3D you use the sine of the angle between the two vectors times the lengths of the vectors, but in 4D there are 3 different angles involved. For an "nd cross product" you need (n-1) vectors. Exercise for the reader, is the determinant of

Unfortunately, the cross product does not carry as nicely into higher dimensional spaces as the inner prodcut did. So in some sense an inner product is a more fundamental concept.

An operation with some similarities to the 3D cross product is the exterior product which is defined on objects called differential forms . In n-space, differential forms come in different dimensions. An m dimensional differential form is called an m-form. If m> n the form is zero, so effectively we must have m< =n.

An m-form acts on m vectors in n-space to give you a scalar (i.e. a real number) and it is antisymmetric in its arguments (so, for example if w is a 2-form and v1 and v2 are vectors then w(v1,v2) = - w(v2,v1)).

The exerior product is defined to act on an m-form and an n-form to give an (m+n)-form. It has the properties:

a ^ b = (-1)^(mn) b ^ a
a ^ a = 0

So it has the same anticommutativity properties as the cross product. Its defination also uses determinants but is a bit complicated.

I don't know if any of the above is intelligible, I can elaborate on forms if you like, but the main point here is that there is something similar to a cross product, but it is substantially more complicated than generalising the inner product.

Sean