Dimension Question: How Does Math Shrink 3-D Shapes?

[Wizardsblade]

I remember when I was in school my professor mathematically shrank a cube to a point and when he did so it was a sphere. So I was wondering does this mean that a sphere is the best way to think 3-D. Then I also wondered what would a square shrink to, I would guess a circle but I do not remember how he did this. Then I thought about a line... would it turn out to be a vector or would it stay a line? Anyhow I would be grateful if someone knew these answers of knows how the math was done to show this.
Thanks

 

[D H]

I remember when I was in school my professor mathematically shrank a cube to a point and when he did so it was a sphere. ... Anyhow I would be grateful if someone knew these answers of knows how the math was done to show this.

A point is a zero-dimensional object. It is not a line segment, a circle, or a sphere, or any other dimensioned object. It is mathematically invalid to speak of a point as being a sphere since a sphere has a non-zero radius by definition. For that matter, it is mathematically invalid to speak of a point as being any dimensioned object. Your professor was playing games with degeneracies.

 

[tacman]

Math can indeed be used to shrink 3-D shapes, and it is a fascinating concept to explore. The process of shrinking a shape involves using mathematical equations and principles to reduce the size of the shape while maintaining its overall structure and properties. In your example, your professor was able to shrink a cube to a point, resulting in a sphere. This is because a point is the smallest possible representation of a 3-D shape, and a sphere is the shape with the most symmetry and equal distance from its center point.

While a sphere may be the most common and intuitive 3-D shape, it is not necessarily the "best" way to think about 3-D. Different shapes have different properties and uses, and it ultimately depends on the context in which the shape is being used. For example, a cube may be a more practical shape for building structures, while a sphere may be more useful for calculations involving volume.

Shrinking a square to a circle is also possible using mathematical equations. This is because a circle is essentially a 2-D version of a sphere, with all points on the edge equidistant from the center. A line, on the other hand, would not become a vector when shrunk, as a vector is a mathematical object with both magnitude and direction. A line would simply become a point when shrunk using mathematical principles.

Overall, the process of mathematically shrinking 3-D shapes is a complex and interesting concept that requires a deep understanding of geometry and mathematical principles. If you are interested in learning more about this topic, I would recommend exploring resources on geometry and spatial mathematics. There are also various online tools and simulations that allow you to visualize and manipulate 3-D shapes using mathematical equations. I hope this helps answer your questions and sparks your curiosity about the wonders of math!