Powers higher than third

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In summary, topology is a field that considers spaces of dimensions higher than three, which may be more complex than traditional rectangular spaces. There are books, such as Steenrod's or Hinton's, that offer methods for visualizing four-dimensional space, but these may be difficult to find. One method involves using colored cubes to represent the faces of a 4D solid, but it is unclear how to reproduce this method. While topology may also cover more advanced concepts like wild topology, simpler spaces like lens space can provide a glimpse into this field.
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scientifico
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Hello, I can graphically image square power as a surface and third power as volume but how can I image powers higher than third ?
 
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Topology routinely considers spaces of dimension higher than three and some of those are much stranger than the "nice rectangular" spaces you might ordinarily think of. You might look at a few topology books and see if you can find one that meets your needs, but topology covers lots of other things and you might have difficulty finding just what you want.

Steenrod, if I haven't gotten him confused with another name, wrote a book in the last twenty years that specifically described one method of visualizing 4 space. Unfortunately my copy is buried so deep that I'll likely never find it again.

Charles Howard Hinton was an individual about a century or a bit more ago who wrote several books on the fourth dimension.

http://en.wikipedia.org/wiki/Charles_howard_hinton

Dover I think published a chopped up version of parts of this about 40 years ago, but some university libraries might have reprints of Hinton's originals. He had what was claimed to be an astonishing memory. In his book he described a set of colored cubes where the colors showed how the faces would be glued together to create a 4D solid if we just had 4 space to work in. It was claimed that if you just worked hard enough, and perhaps had good enough memory, that after a while you would begin to be able to "see" 4 space, either that or you just learned how to quickly correctly answer questions about 4 space as if you could see it. Unfortunately I have never been able to find a precise description of the coloring of the cubes or the complete original instructions so it would be possible to try this. Rudy Rucker in one of his books claims to have been able to reproduce a set of the cubes and followed the instructions, but I do not remember the outcome.

There is a whole field called "wild topology" that is far outside what I expect you are starting with, but lens space is much tamer than that and can give you a peek into the simpler spaces that topology studies.

http://en.wikipedia.org/wiki/Lens_space
 
  • #3
scientifico said:
Hello, I can graphically image square power as a surface and third power as volume but how can I image powers higher than third ?

You don't; you just know that they behave a certain way, algebraically speaking.
 

1. What are powers higher than third?

Powers higher than third refer to mathematical operations involving exponents that are greater than three. These powers can be expressed as x^n, where n is an integer greater than three.

2. What is an example of a power higher than third?

An example of a power higher than third is x^5. This can be expanded as x*x*x*x*x, where there are five x's multiplied together.

3. How do you solve equations with powers higher than third?

To solve equations with powers higher than third, you can use the rules of exponents to simplify the expression. This may involve combining like terms, factoring, or using the power of a power rule.

4. What are some real-world applications of powers higher than third?

Powers higher than third are commonly used in physics and engineering to describe relationships between variables, such as in the equations for rotational motion and electrical circuits. They are also used in finance to calculate compound interest.

5. Can powers higher than third be negative or fractional?

Yes, powers higher than third can be negative or fractional. For example, x^-4 and x^(1/2) are both considered powers higher than third. Negative and fractional powers can be solved using the rules of exponents and may result in irrational numbers.

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