Mathematical objects, structures and spaces

[autodidude]

What are they, how are they visualized? Is it like how basic equations when plotted form a shape (e.g. circle) but much more complex?

When I think of the those words, I think of some sort of weird/cool looking shape.

 

[TGlad]

Can mean a lot of things, but commonly mathematical objects are defined as sets. For example a sphere can be defined as the set of points x,y,z such that the sum of the squares equals 1.
That is shorthand for {(1,0,0), (0.999, 0.001, 0.001), (0.998, 0.003, 0.003), ...}
In fact it is an infinite set of points... which in this case forms a surface.

The mandelbrot set is similarly just a set of 2d points, or rather, complex numbers.

Sometimes you don't even specify the set of points, they are just concepts. For example the mobius strip is a ribbon where the end is attached to the start but rotated 180 degrees. The exact shape or length of the strip is not important.

Did that help?

 

[chiro]

What are they, how are they visualized? Is it like how basic equations when plotted form a shape (e.g. circle) but much more complex?

When I think of the those words, I think of some sort of weird/cool looking shape.

Hey autodidude.

It depends. You could be describing an object like a surface, volume or other higher dimensional object and in one way, you could graph the actual surface/whatever itself (and also projections if it is a higher dimensional object), or you could use a 2D venn diagram that is transformed in a way that makes more graphical intuitive sense.

Usually when you think about probability, it can make sense to use a Venn diagram, or maybe to use several Venn diagrams in some scenarios.

What you will find is that depending on what you are talking about, new kinds of representations whether they are algebraic or geometric creep in. You'll see this anywhere, whether it is algebra (graduate algebra not high school algebra), or something else.

But yeah sometimes it can be useful to reduce objects down to points and then represent those points in venn diagrams. There is a lot of transformation and work going on behind the scenes in this regard, but it can help make sense of something that is otherwise more complex.

 

[HallsofIvy]

Everything in mathematics is an "object", "structure", or "shape"!

 

[autodidude]

@TGlad: yeah, that makes some sense to me

thanks guys

 

[Number Nine]

When mathematicians talk about "structures" and "spaces", they're generally talking about collections of objects with rules governing how they behave and interact.