Problem in Higher Dimensions

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

more…

CrazyNeutrino said:

Problem in Higher Dimensions

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

more…

I bet the 24 dimension result is the Leach lattice. Am I right?

mollwollfumble said:

CrazyNeutrino said:

Problem in Higher Dimensions

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

more…

I bet the 24 dimension result is the Leach lattice. Am I right?

—-

Can you first instruct us into understanding the fifth to the 24th dimension?

tauto said:

mollwollfumble said:

CrazyNeutrino said:

Problem in Higher Dimensions

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

more…

I bet the 24 dimension result is the Leach lattice. Am I right?

—-

Can you first instruct us into understanding the fifth to the 24th dimension?

I’ve already written a guide to the fourth dimension, aimed at final year high school students. I’ll dig it out. The only three things you need to know to work in higher dimensions is how to write coordinates, how to calculate distances, and how to calculate dihedral angles. Another thing that helps, you know there are 5 Platonic solids in 3-D, well in 5 and more dimensions there are always exactly 3 Platonic solids, one is analogous to the tetrahedron, one to the cube, and one to the octahedron.

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

Cymek said:

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

what sort of effect do you mean?

diddly-squat said:

Cymek said:

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

what sort of effect do you mean?

I am wondering if they actually serve a purpose, are they like the coding for the universe and you change a higher dimension as it changes a constant in the 4 dimensional universe, sounds far fetched but I wonder what they do.

Cymek said:

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

Presumably string theorists would say they do.

I resolved a TOE but I was out of my gourd at the time and forgot to write it down.

Cymek said:

diddly-squat said:

Cymek said:

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

what sort of effect do you mean?

I am wondering if they actually serve a purpose, are they like the coding for the universe and you change a higher dimension as it changes a constant in the 4 dimensional universe, sounds far fetched but I wonder what they do.

it’s a coordinate system, the only purpose it serves is as a means to make particular types of mathematics more elegant.

diddly-squat said:

Cymek said:

diddly-squat said:

what sort of effect do you mean?

I am wondering if they actually serve a purpose, are they like the coding for the universe and you change a higher dimension as it changes a constant in the 4 dimensional universe, sounds far fetched but I wonder what they do.

it’s a coordinate system, the only purpose it serves is as a means to make particular types of mathematics more elegant.

At some point in time though did it serve a purpose and now they are just a dead remnant from the early universe were things were still forming.

Cymek said:

diddly-squat said:

Cymek said:

I am wondering if they actually serve a purpose, are they like the coding for the universe and you change a higher dimension as it changes a constant in the 4 dimensional universe, sounds far fetched but I wonder what they do.

it’s a coordinate system, the only purpose it serves is as a means to make particular types of mathematics more elegant.

At some point in time though did it serve a purpose and now they are just a dead remnant from the early universe were things were still forming.

I’m not sure I understand; it’s only a mathematical thing… we use these models because they are particularly elegant ways to describe things like, for instance, high energy particle physics.

diddly-squat said:

Cymek said:

diddly-squat said:

it’s a coordinate system, the only purpose it serves is as a means to make particular types of mathematics more elegant.

At some point in time though did it serve a purpose and now they are just a dead remnant from the early universe were things were still forming.

I’m not sure I understand; it’s only a mathematical thing… we use these models because they are particularly elegant ways to describe things like, for instance, high energy particle physics.

So the higher dimensions only exist in mathematics not in reality ? or not in human reality as we can only exist in four dimensions

Cymek said:

diddly-squat said:

Cymek said:

At some point in time though did it serve a purpose and now they are just a dead remnant from the early universe were things were still forming.

I’m not sure I understand; it’s only a mathematical thing… we use these models because they are particularly elegant ways to describe things like, for instance, high energy particle physics.

So the higher dimensions only exist in mathematics not in reality ? or not in human reality as we can only exist in four dimensions

all dimensions are mathematical only… they are used to build up a mathematical model of a physical reality.

diddly-squat said:

Cymek said:

diddly-squat said:

I’m not sure I understand; it’s only a mathematical thing… we use these models because they are particularly elegant ways to describe things like, for instance, high energy particle physics.

So the higher dimensions only exist in mathematics not in reality ? or not in human reality as we can only exist in four dimensions

all dimensions are mathematical only… they are used to build up a mathematical model of a physical reality.

True but we interact with four of them, could we access these higher dimension I wonder as a means to manipulate fundamental forces in our universe.

Cymek said:

diddly-squat said:

Cymek said:

So the higher dimensions only exist in mathematics not in reality ? or not in human reality as we can only exist in four dimensions

all dimensions are mathematical only… they are used to build up a mathematical model of a physical reality.

True but we interact with four of them, could we access these higher dimension I wonder as a means to manipulate fundamental forces in our universe.

I’m not convinced that we interact with the three spatial and one time dimension in any meaningful physical way. They are simply abstract concepts used to describe a physical world.

How do you interact with length or breadth or height?

diddly-squat said:

How do you interact with length or breadth or height?

If you go for a walk, you are interacting with length.
If you turn through a right angle and walk some more, you are then interacting with breadth.
If you then turn through 45 degrees and walk up some steps you are interacting with all three.

You are interacting with time all the time.

So the question is, are there further dimensions that have a physical effect on the way things behave?

The Rev Dodgson said:

diddly-squat said:

How do you interact with length or breadth or height?

If you go for a walk, you are interacting with length.
If you turn through a right angle and walk some more, you are then interacting with breadth.
If you then turn through 45 degrees and walk up some steps you are interacting with all three.

You are interacting with time all the time.

So the question is, are there further dimensions that have a physical effect on the way things behave?

you are interacting with solid objects, not the dimensions; it just so happens that these same objects are well described by 3 dimensional space

Cymek said:

Do these higher dimension have any direct effect on the 4 dimensions we are familiar with.

“Dimension” doesn’t have to refer to physical space. Temperature is a perfectly valid dimension, at any given time every point in space has three space dimensions and one temperature dimension. Solving Newton’s equation by working in “phase space” involves solving equations in six dimensions at every timestep, three dimensions for location and three more dimensions for velocity/momentum. Quite recently I was solving a quantum mechanics problem in seven dimensions, the dimensions came from the discretisation of the path integral so, for instance, in order to do the solution I needed to use the 6-D surface area of a 7-D hypersphere. In simulated annealing problems the number of interdependent dimensions can be in the thousands.

tauto said:

mollwollfumble said:

CrazyNeutrino said:

Problem in Higher Dimensions

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

more…

I bet the 24 dimension result is the Leach lattice. Am I right?

—-

Can you first instruct us into understanding the fifth to the 24th dimension?

You asked for it.

I’ve assembled a crash course on understanding n-D geometry in this document http://freepages.misc.rootsweb.com/~hallsofjamaica/Crash%20Course%20in%20nD.pdf

There are three short chapters. The first is on calculating lengths and angles in n-D. The second is on regular polytopes (polygons and polyhedra are examples of polytopes) in n-D. The third is on n-sphere packings in n-D. Enjoy.

mollwollfumble said:

tauto said:

mollwollfumble said:

I bet the 24 dimension result is the Leach lattice. Am I right?

—-

Can you first instruct us into understanding the fifth to the 24th dimension?

You asked for it.

I’ve assembled a crash course on understanding n-D geometry in this document http://freepages.misc.rootsweb.com/~hallsofjamaica/Crash%20Course%20in%20nD.pdf

There are three short chapters. The first is on calculating lengths and angles in n-D. The second is on regular polytopes (polygons and polyhedra are examples of polytopes) in n-D. The third is on n-sphere packings in n-D. Enjoy.

—-

Thanks. Now I remember why I forgot 3U maths. Too hard unless you use it regularly.

diddly-squat said:

The Rev Dodgson said:

diddly-squat said:

How do you interact with length or breadth or height?

If you go for a walk, you are interacting with length.
If you turn through a right angle and walk some more, you are then interacting with breadth.
If you then turn through 45 degrees and walk up some steps you are interacting with all three.

You are interacting with time all the time.

So the question is, are there further dimensions that have a physical effect on the way things behave?

you are interacting with solid objects, not the dimensions; it just so happens that these same objects are well described by 3 dimensional space

I am interacting with solid (and gaseous and liquid) objects in a way that involves the specified dimensions.

Why do you not see this as interacting with the dimensions?

The point is that space dimensions have specific properties that other things that we can assign to dimensions do not have, so it makes sense to ask whether there are other spatial dimensions.