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An Infinitely Large Napkin

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Re: An Infinitely Large Napkin

#21

Post by derefr »

todd8 wrote: post #14

I'm a computer scientist, not a mathematician (but I've taken around 25 college math courses spread over many years). Nevertheless, I do have a couple of suggestions and observations that I will address to the author that I hope is seeing this. First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for…

The point of the "napkin" isn't to be a generic Maths textbook; it's to trace up the prerequisite chain from category theory until it connects with high-school-level maths. Do you need statistics or differential equations to understand category theory?

Re: An Infinitely Large Napkin

#22

Post by improv32 »

berbec wrote: post #20
LeonB wrote: post #19

If there was an infinitely large napkin, floating in space for example, far away from everything else, would its own gravity cause it to crumple toward any folds/creases/imperfections, with sufficient force to create a cascading implosion of sufficient mass/density to create a black hole?

An infinitely large napkin, if made of any sort of normal matter, would have infinite mass. It's gravitational pull would propigate, at c IIRC, out from the instant of its creation across all of space/time, sucking all objects (including itself) towards it's center of mass at the speed of light (OK, just under. Literally c-0.000...1m/s). It not only would create a black hole, it would signal the end of the entire uni…

How can an infinitely large napkin have a center of mass? It seems to me that all everything would get sucked on a trajectory normal to the napkin's surface. Any horizontal force would get cancelled by an opposite force, at an opposite point on the napkin. Also, force of gravity drops off with the inverse square of distance, so the force would have a finite value, depending on the density of the napkin. Similar idea to a common homework problem for introducing electrostatics: calculate the electric field generated at some point by an infinitely long wire with some charge density.

Re: An Infinitely Large Napkin

#23

Post by berbec »

improv32 wrote: post #22
berbec wrote: post #20

Earlier quoted context omitted.

An infinitely large napkin, if made of any sort of normal matter, would have infinite mass. It's gravitational pull would propigate, at c IIRC, out from the instant of its creation across all of space/time, sucking all objects (including itself) towards it's center of mass at the speed of light (OK, just under. Literally c-0.000...1m/s). It not only would create a black hole, it would signal the end of the entire uni…

How can an infinitely large napkin have a center of mass? It seems to me that all everything would get sucked on a trajectory normal to the napkin's surface. Any horizontal force would get cancelled by an opposite force, at an opposite point on the napkin. Also, force of gravity drops off with the inverse square of distance, so the force would have a finite value, depending on the density of the napkin. Similar idea…

True if the napkin is perfectly flat. Any imperfections would negate the cancelation of opposing force, no?

The drop off would only apply if the napkin has finite mass or infinite size, correct?

Re: An Infinitely Large Napkin

#24

Post by improv32 »

berbec wrote: post #23
improv32 wrote: post #22

Earlier quoted context omitted.

How can an infinitely large napkin have a center of mass? It seems to me that all everything would get sucked on a trajectory normal to the napkin's surface. Any horizontal force would get cancelled by an opposite force, at an opposite point on the napkin. Also, force of gravity drops off with the inverse square of distance, so the force would have a finite value, depending on the density of the napkin. Similar idea…

True if the napkin is perfectly flat. Any imperfections would negate the cancelation of opposing force, no? The drop off would only apply if the napkin has finite mass or infinite size, correct?

Indeed, if it wasn't flat it would not perfectly cancel.

The drop off applies because the force of gravity is proportional to 1 / distance^2, so in this case the napkin would still have infinite mass, but it would not have infinite density, so if you took a surface integral of the gravitational force provided by each point in the napkin, over the whole napkin, it would converge on a finite value, as each point contributes less and less force as you get father away.

Re: An Infinitely Large Napkin

#25

Post by improv32 »

berbec wrote: post #23
improv32 wrote: post #22

Earlier quoted context omitted.

How can an infinitely large napkin have a center of mass? It seems to me that all everything would get sucked on a trajectory normal to the napkin's surface. Any horizontal force would get cancelled by an opposite force, at an opposite point on the napkin. Also, force of gravity drops off with the inverse square of distance, so the force would have a finite value, depending on the density of the napkin. Similar idea…

True if the napkin is perfectly flat. Any imperfections would negate the cancelation of opposing force, no? The drop off would only apply if the napkin has finite mass or infinite size, correct?

Here's a cool link I just found going over this exact scenario: https://www.mathpages.com/home/kmath530/kmath530.htm

Interesting to note, the napkin creates a uniform gravitational field above and below it. Meaning that the force applied to an object is the same regardless of how far away it is from the napkin! That force is 2pi * G * m * rho where rho is the mass density of the napkin.

Re: An Infinitely Large Napkin

#27

Post by uptownfunk »

Was not at all what I was expecting from the title. Love the work, (of course I majored in math at Uni).

I totally support and encourage any efforts to make higher math more approachable and understandable. I remember the multiple hazings I went through with Rudin (both little, big, and functional analysis). The comic in the beginning is hilarious.

Re: An Infinitely Large Napkin

#28

Post by joe-collins »

I've got a few years of community college calculus under my belt, but the cost of university and the unstructured nature of MOOCs had both deterred me from advancing along those lines. Wikipedia's math pages typically pre-suppose a higher level of understanding than I possess, or are written in exquisitely correct but practically impenetrable fashion.

This looks like it may be the bridge I've been seeking. Thank you.

Re: An Infinitely Large Napkin

#29

Post by todd8 »

derefr wrote: post #21
todd8 wrote: post #14

I'm a computer scientist, not a mathematician (but I've taken around 25 college math courses spread over many years). Nevertheless, I do have a couple of suggestions and observations that I will address to the author that I hope is seeing this. First, it would be a very sophisticated high school student to tackle topology and some of the other areas of abstract mathematics. I really like the topics you've picked for…

The point of the "napkin" isn't to be a generic Maths textbook; it's to trace up the prerequisite chain from category theory until it connects with high-school-level maths. Do you need statistics or differential equations to understand category theory?

I like the book but the contents seem to go well beyond the prerequisites of category theory—-it’s almost 900 pages.

I got the impression that the author was not simply attempting to connect high school math to category theory but was providing a broader survey of higher math. I interpreted the author’s remarks about the path to category theory as the inspiration for embarking on the project that has turned out to be a wide survey of higher math that might benefit young mathematicians.

Re: An Infinitely Large Napkin

#30

Post by axiom92 »

throwawaymath wrote: post #2

Adding this here as it may be of related interest for those who enjoyed the massive math cheat sheet on the front page recently. Evan Chen, a math student at MIT, wrote up what would be considered field notes for higher mathematics. The full PDF is here[1], complete with a dependency graph showing what you need to know before reading any particular section. ____________ 1. https://usamo.files.wordpress.com/2019/02/na…

Am I missing something here? The link you've shared is exactly what the post is about.
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