The True Basic Vectors Of The Third Dimension And The Values Of Other Dimensions

[SketchTurner]

Ok, this might take a couple of replies for me to put this in a way people can understand but here's a first shot at it.

Last night I was about to go to sleep when I had a strange thought - Why do people view the universe as three perpendicular vectors (up/down, left/right, back/forth). Also similarly the 2D universe would be viewed and measured according to a square grid. But why cubes and squares? Sure they seem an obvious way to look at things with the nice right angles but when trying to link the 2D to the 3D it makes things a lot more complex. For example: Take a square. Simple 2D object, four corners, four edges, one face and of course being 2D it’s completely flat. However because there are four corners it is possible to distort the shape three dimensionally making it no longer a 2D object. (If you don’t know what I mean just take a square of card, hole 3 corners flat to a table and pull the other up. It folds an you now have two triangles)

So my idea was that the square was not an easy shape to use as a basis for investigating patterns in different dimensions. A much better shape is the triangle. It’s the simplest possible 2D shape therefore making it the easiest to compare across dimensions. I’ll explain – The triangle is the simplest 2D shape just as a pyramid is the simplest 3D shape. They both have the least possible number of corners and edges for their dimension.

So what about the first dimension? Well the equivalent would be a single line with two points and a connecting line. This may seem the simplest possible conceivable as it is one line however to investigate number patterns I had to use two lower dimensions – 0D (a single point with no dimensions infinitely small) and “No D” where there is nothing.

 

Right here’s the explanation:

I will use the term “points” to describe endpoints of a line or corners of a shape. I will use the term “lines” to describe the lines between two points or the edges of a shape. I will use the term “faces” to describe areas enclosed by three lines (triangles). I will use the term “volumes” to describe 3 dimensional objects (in this case all will be tetrahedrons which I’ll call pyramids because it’s easy to type). “4ths and 5ths” will be the fourth and fifth dimensional equivalents.

To keep with the rules of the lines, triangles and pyramids, the 4D and 5D shapes must be the simplest possible in their dimension and every point must connect to every other point in the shape.

 

Starting with “No D” there are no points therefore no lines, faces and so on.

In 0D there is 1 point but no lines, faces etc.

In 1D there are 2 points and 1 connecting line but no faces or volumes

In 2D there are 3 points, 3 lines and one face but no volumes

In 3D there are 4 points, 6 lines, 4 faces and 1 volume

Here’s where the patterns come in. It’s also where it’s a lot easier to explain with a model but I’ll try. My theory was that according to a pattern emerging a 4D shape would have to be made of 5 points, 10 lines, 10 faces and 5 volumes. I simply too a bunch of toothpicks and some plasticine and made a model of a pyramid. To make the 3D model of the 4D shape I added another ‘point’ in the centre and connected it to all the other points in the pyramid. The result was a shape consisting of 5 points, 10 lines, 10 faces and 5 volumes (if you make the model you can count 4 pyramids using the central point and 1 using only the outer points.

Ok, time for a spreadsheet. First I drew a rough table and worked out the rules. Simple to figure and obvious now I think about it but the number of points from 0D onwards are triangular numbers. The number of lines (starting at 1D) goes up in pyramid numbers. I simply took the relationship between integers, triangle numbers and pyramid numbers and extended it with a simple equation. I've included the table and a small example of it with the equation showing so you can play around with it.

 

That’s about where it ends, it’s only been a few hours since I thought of it so you can’t expect much. I wouldn’t be surprised if somebody had thought of this before but I thought I’d share it anyway. I’ll answer anything you don’t understand about this (except for why I was thinking about triangles in bed).

okspreds.doc

[DQW]

The reason people like rectangular co-ordinates (or spherical, cylindrical, etc.) is that it is useful to have orthonormality between basis vectors, and the dot product is a convenient inner product.

[Klaynos]

So, forgive me if I've got you wrong, but you are trying to form a new coordinate set instead of the standard cartessian x,y,z set?

 

I simply too a bunch of toothpicks and some plasticine and made a model of a pyramid. To make the 3D model of the 4D shape I added another ‘point’ in the centre and connected it to all the other points in the pyramid.

 

Surely this is trying to generare (3+n)Dimention shapes in 3D space which is very very difficult at the best of times?

 

Have you ever looked at polar coordinate sets?

 

[math](\rho,\theta,\phi)[/math]

 

etc...

[SketchTurner]

Ok forget the title for now obviously it's leading people away from what I actually did. I was just saying that it's easier to investigate triangles and tetrahedrons than squares and cubes.

About the 4D shape, I made it, looked at it and with a little thinking it's not too hard to comprehend how it would work

[SketchTurner]

Thought I'd add that looking at a 4D shape in 3D is like trying to see a picture of a 3D shape drawn on a piece of paper. Obviously it isn't 3D but you can make the picture 'pop out' of the page with your mind. To understand 4D you have to view the 3D representation from all sides and then 'pop it out' in your mind. Hard to explain really but after looking at the model I found it quite easy.