Volume of a 4 dimensional orb equation

[BlackJack]

In a lecture today, we (my professor) calculated the volume of a 4 dimensional orb. I discussed it with a few other people but no one could properly explain how to picture this orb.

What do you think does it "look" like?

One possible theory: An orb which depends on the time and shiftes it's shape accordingly. But this doesn't seem satisfactory to me.


Help ? :uhh:

 

[Moo Of Doom]

You CAN'T imagine what a 4D orb "looks" like. I can think of no PhD who has been working with higher dimensions for a long time who would admit to being able to picture a 4D object. The closest thing you can get, pictorally, is to slice the object into 3D slices and associate each slice with a different 4th coordinate. In that case, a 4D orb starts at a point at (0,0,0,-r) grows in a circular fashion as you move towards 0 along the 4th axis until it forms the sphere x^2+y^2+z^2=r^2 centered at (0,0,0,0) and then shrinks again in a circular fashion until you hit (0,0,0,r). The radius of the 3D spherical slice at the 4th coordinate (let's say w) w=a where -r<a<r is going to be sqrt(r^2-a^2). That's pretty much what a 4-sphere is.

 

[tacman]

The concept of a 4 dimensional orb can be difficult to visualize, as our brains are only able to perceive and understand objects in three dimensions. However, we can try to understand it through mathematical equations and analogies.

Firstly, it's important to understand that a 4 dimensional orb is a theoretical concept and cannot physically exist in our three-dimensional world. It is a mathematical construct used by physicists and mathematicians to study higher dimensions and complex systems.

To visualize a 4 dimensional orb, imagine a regular 3 dimensional sphere. Now, add an extra dimension to it - a fourth dimension. This fourth dimension can be represented by time, as you mentioned, but it can also be represented by other parameters such as temperature or energy.

In this 4 dimensional space, the orb can change its shape and size over time or as the other parameters vary. It can expand, contract, twist, and morph in ways that are impossible for us to imagine in our three-dimensional world.

Another way to think about it is through analogy. Just like how a 3 dimensional sphere can cast a 2 dimensional shadow, a 4 dimensional orb can cast a 3 dimensional "shadow" in our world. This shadow would appear as a constantly changing and evolving object, similar to the way a 3 dimensional object would appear to a 2 dimensional being.

In terms of the equation for the volume of a 4 dimensional orb, it would involve four variables - three for the three spatial dimensions and one for the fourth dimension. This equation would be complex and difficult to visualize, but it can be solved using mathematical techniques.

In conclusion, while it may be challenging to fully comprehend and picture a 4 dimensional orb, we can use mathematical concepts and analogies to better understand its properties and behavior. The key is to keep an open mind and approach it with curiosity and a willingness to learn. I hope this helps in your understanding of this fascinating concept.