Can you do the math for me regarding a multidimensional universe?

Question

When a 2-dimensional cone passes through a 2-dimensional plane, the result is a single expanding circle. So, here's my question: Will a 3-dimensional cone passing through a 3-dimential space result in a 3-dimensional expanding "universe"?

I am not referring to projections... and of course a cone can be 2-dimensional, in the same sense that the surface of a torus is 2-dimentinonal. Maybe omitting the word "surface" caused the confusion?

spelling is not my strong suit, obviously

Answer 1

I think you mean when a three dimensional cone passes through a 2 dimensional plane then the result is a single expanding circle PROVIDED the base of the cone is parallel to the 2 dimensional plane.

a 3d cone passing through a 3d space results in a 3d cone traveling through space. expansion of the universe in not a necessity of this scenario.


read warped passages by Lisa Randall again.

Answer 2

There is, of course, no such thing as a "2-dimensional cone." By definition, a cone is 3D.

What you are really saying, is when the projection of a 3-dimensional cone falls on a 2-dimensional plane.... Totally different ballgame from what you described.

Like the name implies "projection" conjures the image of a bright light shining on a 3D object so that the object's shadow is projected onto a screen, the 2D plane. And, clearly, as you bring the cone closer to the screen, its shadow will expand until its 2D dimensions on the screen are approximately equal to the cross sectional area dimensions of the 3D cone. And if the cone is projected end-on (either end), the projected image, the shadow, will be a circle.

Curvilinear coordinates is a subset of vectors analysis. This subset focuses on issues like you brought up with your question. That is, how do higher dimensions project onto lower dimensions? It also addresses how to choose the right dimensional metrics (e.g., Cartesean vs. spherical coordinates) to make problem solving easier.

A major motivator for the dimensional focus is that we are finding more and more physics theories that posit dimensions higher than our usual four (3D plus time). As you might well imagine, solutions in higher dimensions are difficult if not impossible to come by. So they, the physicists and their math buddies, try to project those higher dimensions onto lower ones so solutions can be found.

So, to answer your question, a 5D object projected onto our 4D universe would appear as a 4D object. That is, the projection reduces the dimensions by one. And that's a general rule, projection reduces the original dimensions by one. But, and this is a big BUT, we can do more than one projection of an object.

For example, we can project your 3D cone onto a 2D screen. Then we can project the 2D shadow onto a 1D line. Vectors analysis is a very powerful tool in the toolshed of physicists and mathematicians.