Math Is Fun Forum

random_fruit

Member

Registered: 2008-12-25

Posts: 39

Three dimension maths

One could say the real number line is one dimensional.  Then, one could say that imaginary numbers give us a second dimension, and complex numbers allow us to 'explore' the plane.  So, what piece of maths allows us to 'move' into a three-dimensional space?

I know we could just give 3 real numbers (x,y,z) which describe a point in 3D.  I could for myself work out 3 separate lines/planes/solids using 3 sets of one-dimensional co-ordinates.  But, that NOT what I want.  What I wish is an idea as 'clever' or as 'complicated' or even as 'simple' as imaginary numbers which takes us from two to three dimensions.

Has this ever been done?  I so, I have yet to hear of it.  Mind you, that's not to say either way... 'cos my maths education pretty much stopped at age 18 before university.  But I am curious that as we live in a 3D world I have not yet come across a 3D maths.

Can an answer be put in 'simple' terms, such as I can recall from 1975 when I finished 'A' level maths in the UK?  Thanks,

random_fruit

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Three dimension maths

There's no similar way to pull a third dimension from the complex numbers, because the complex numbers are algebraically closed.

In the reals, x² + 1 is made of real coefficients but has no real roots. The imaginary number i can then be defined as a root of this polynomial, and we then get complex numbers by making linear combinations of 1 and i.

However, given a polynomial

, with complex coefficients, this will always have n complex roots. There are no "gaps" in this number system and so no third dimension for it to go to.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Three dimension maths

There's no similar way to pull a third dimension from the complex numbers, because the complex numbers are algebraically closed.

It is important to keep in mind that this is one way to look at the complex numbers, but not the only way.  The complex numbers form a 2 dimensional vector space over the real numbers, with multiplication defined in a funny way.  They actually have a lot more properties than that, but this is just one view.

The question is can we do the same thing with a third component, extending the complex numbers?  The answer is a rather surprising no.  We would have to lose a fundamental algebraic property to do so (I think it was the associative property, but I need to look this up).  However, as discovered by Hamilton, we can do such a thing with four components, and these are known as the Quaternions.  And so we have 1, 2, 4, and the next number is the Octonions.

http://en.wikipedia.org/wiki/Quaternion
http://en.wikipedia.org/wiki/Octonion

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

random_fruit

Member

Registered: 2008-12-25

Posts: 39

Re: Three dimension maths

Thank you both.  The words "algebraically closed" tell me what I needed to know, but I hadn't made the leap that rules out 3D.  when I have some time to spare I think I had better look Quaternions up in Wiki and hope I can understand some of that.

random fruit

Last edited by random_fruit (2009-05-17 07:05:48)

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

Ok, but what would happen if you made a j so that sqrt(negative sqrt(-1)) = sqrt(-i) = j? Would there be any contradictions?

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 160

Re: Three dimension maths

Very nice thought... j^4 = j^2 * j^2 = (-i) * (-i) = -1
In this case, a 3D complex number could be expressed as: A + Bi + Cj
This could lead us to 4D complex numbers as: A + Bi + Cj +Dk where k^8=-1
and so on...

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

YES! EXACTLY WHAT I WAS THINKING! Thank you!

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

But there's still the problem that there might be contradictions.

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

Uhhhh, no, not quaternions, those are defined by i^2=j^2=k^2=ijk=-1. Me and KerimF meant i^2 = j^4 = k^8 = -1.

Robert-not-Smart-enough

Novice

Registered: 2024-04-08

Posts: 2

Re: Three dimension maths

The mathematical concept that extends our understanding from two dimensions to three dimensions is quaternions. Quaternions are an extension of complex numbers and provide a way to represent rotations and orientations in three-dimensional space.

Unlike complex numbers, which have a real part and an imaginary part, quaternions have four components: one real part and three imaginary parts. They are often written in the form:

q=a+bi+cj+dk

where a,b,c, and d are real numbers, and i,j, and k are imaginary units that satisfy certain multiplication rules.

Quaternions are particularly useful in computer graphics, robotics, and aerospace engineering for representing rotations and orientations because they avoid the problems associated with other representations like Euler angles. They provide a concise and efficient way to describe 3D rotations, making them a powerful tool in three-dimensional mathematics.

In working through this mathematical problem, it's clear that precision and accuracy are key. For those seeking assistance or guidance with such calculations, I will suggest you to try spam.com that can be incredibly valuable. Also, you can contact them at more spam.

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

You're not getting it. I didn't mean quarternions.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,136

Re: Three dimension maths

hi SONIC123CDMANIA

Robert-not-Smart-enough is just a spammer trying to get free advertising for a "we'll do your maths assignment for you ... just pay us lots of money" company.

They have tried 7 times using the following method:

(1) join
(2) pick a random topic and use AI to make a 'relevent' comment
(3) conclude with an advert.

The first 5 times I deleted the member and all posts.

For the latest two I have experimented with two new approaches.

In http://www.mathisfunforum.com/viewtopic … 76#p439076  I 'unadvertised them' by explaining why I think it wise not to use their services.  Have a look at the link if you want a laugh.

In this post I set up censorship rules so the advert is automatically spam-ed out.

So far they don't seem to have noticed, but they have, so far, stopped making new posts.

It doesn't say much for their maths skills that they didn't even understand the topic.

Meanwhile back to your question.

Ok, but what would happen if you made a j so that sqrt(negative sqrt(-1)) = sqrt(-i) = j?

The set of complex numbers is closed for all algebraic equations. This should mean that sqrt(-i) should have a complex value.

I did some Argand diagram work and came up with

Check:

There is a second complex root too. I'll leave that as an exercise.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

I'm just asking if there would be any contradictions.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,136

Re: Three dimension maths

You said

Me and KerimF meant i^2 = j^4 = k^8 = -1

By the closure property i, j, and k are all complex numbers ie. can be written as a + bi, so the above doesn't introduce any new 'dimensions' of numbers.

Quaternions (sorry to re-introduce them) are defined in terms of a 4 dimensional vector space, with rules that allow mathematicians to 'multiply' them.

It's possible you could do the same with a 3 dimensional vector space, but you'd have to develop the 'rules' and, in particular, show that your numbers behave in a consistent way.  Is that what you mean by contradictions?  To explore that you need to introduce a full set of rules (axioms) for how you want your numbers to behave.

For example: what is ij ? When I know how to 'multiply' any pair of your numbers I can begin to try and answer your question.

Ricky said

The question is can we do the same thing with a third component, extending the complex numbers?  The answer is a rather surprising no.  We would have to lose a fundamental algebraic property to do so (I think it was the associative property, but I need to look this up).

I've not met this idea before so I cannot confirm or deny this.  I'll do some research.

Later edit:

Google gave loads of links to answer this question. Here's a small subset:

https://tomrocksmaths.com/2022/01/24/st … ks%20down.

"The answer is that such a system is impossible to exist, at least if we want it to have certain ‘nice’ properties. Hamilton couldn’t find a way to divide his 3D numbers and it was later proven that this is exactly the point where a possible three dimensional extension of the complex numbers breaks down. "

https://math.stackexchange.com/question … dimensions

Loads here. The following quote sums it up:

"Consider that whatever system you construct will form a vector space not only over the real numbers, but over the complex numbers as well, and a complex vector space of dimension n  is a real vector space of dimension 2n, so an odd number of dimensions is impossible."

The following video gives us the proof. It's over 36 minutes long and I only skipped through it.

https://www.youtube.com/watch?v=L-3AbJM-o0g

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

Ok, I guess nvm.

Sorry for the stupid question.

Last edited by SONIC123CDMANIA (2024-04-12 00:34:00)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,136

Re: Three dimension maths

Not stupid at all.  We make progress by asking questions and lots of mathematicians have asked this before you.  I didn't know the answer so I had to look it up, so I'm grateful that you have led me to some new knowledge.

Keep on asking!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SONIC123CDMANIA

Member

Registered: 2023-12-15

Posts: 12

Re: Three dimension maths

No, it was a stupid question. I should've known. That's on me.