Things whose inside and outside are ... well ... screwy, verkakte / does not compute / well, actually it DOES compute, but oddly

Click images to enlarge.


The Modern Hiawatha

by George A. Strong
 
 He killed the noble Mudjokivis.
 Of the skin he made him mittens,
 Made them with the fur side inside,
 Made them with the skin side outside.
 He, to get the warm side inside,
 Put the inside skin side outside.
 He, to get the cold side outside,
 Put the warm side fur side inside.
 That's why he put the fur side inside,
 Why he put the skin side outside,
 Why he turned them inside outside. 

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There are a surprising number of Things which really and truly do Exist -- but we mere pathetic humans can never have direct touchy-feely experience with or knowledge of them. These are the Platonic Objects -- objects which can be perceived only through the mind and imagination.

I made and sent Pi Day t-shirts to my lovely, bright grand-nieces in Vermont, and Dad e-mailed me back about the Wonderful Holiday, and the Wonderful Mathematical irrational and transcendental Constant that just makes everyone so happy and makes you want to eat Pi(e).

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[Dear Unkie Munkie]

[The Pi Day t-shirts] have gotten here in time, and they're delightful! We'll have to see whether we have time and/or ingredients for making a pie. The school principal will love these too -- I think he was a math teacher at the high school previously, and I know he's up on math trivia. He recently corrected me when I said that 1 raised to the power of "infinity and beyond" (a favorite "number" of N***'s) = 1. He claims that when you get to infinite powers, weird stuff starts happening.

Love,
Kwak

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NOTE: Dad is one of 3 brothers, and because I could never get their names straight I called  them Huey, Dewey and Louie, and then, after I learned the Dutch names for them, they call themselves Kwik, Kwak en Kwek. (Oddly enough, THEY all knew which brother was which. There's an unconfirmed rumor that you can tell the duckling triplets apart by the color of their baseball caps.)
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EXTREMELY weird stuff starts happening. This sort of quonset hut kind of thing [see lowest illustration above] ... the height of the roof keeps diminishing as it goes to the right ... out to infinity ... so the roof gets closer and closer to the floor, but never quite gets to the floor.

If you want to fill the quonset hut with olive oil, it will store a FINITE volume of olive oil ...

But if you want to PAINT the outside of the hut, you can never buy enough gallon paint cans, because it has an INFINITE surface area. Technically it's a solid of revolution that has a finite volume but an infinite surface area.

And that's very weird. You can fill it, and then go home, but you can't ever finish painting it.

Hope A & N like the shirts! Happy Pi Day! (GAC is off at her sister's in Worcester, but she's eating a Pizza Pi(e)!)

Unkie M.

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That is very cool. I never learned that finite inside/infinite outside thing. Kind of the reverse of the C.S. Lewis's Wardrobe [the Narnia books], which was infinite on the inside yet finite on the outside.

One fun thing I did learn in my precalc course was of one thing that can go faster than light. Imagine that you have a wall going off into the distance, more or less infinitely. It's a straight, flat wall. No bends or curves. Now stand near the wall and point a flashlight at it. Now start to rotate at a fixed speed, so that the spot of light from the flashlight slides across the wall. Because the angle of incidence is changing more rapidly than your speed of rotation, the position of the spot of light on the wall will accelerate exponentially, eventually to beyond-light speed before you rotate enough for the light to be off the wall entirely. Or something like that.

Also, that you can always, ALWAYS, find a position that allows a four-legged table to stand in a stable position on an uneven ground. There's no guarantee it'll be level, but there is a guarantee that it'll be stable. I find that to be a very optimism-inducing idea.

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Don't forget the TARDIS (Time And Relative Dimension In Space) -- on the outside, a sidewalk police phone box. On the inside -- amphitheaters, swimming pools, libraries, laboratories, big closets, kitchen and dining room, etc.

from some nerd chatter on a math nerd site ...

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Infinite area under a curve has finite volume of revolution?

So I was thinking about the harmonic series, and how it diverges, even though every subsequent term tends toward zero. That meant that its integral from 1 to infinity should also diverge, but would the volume of revolution also diverge (for the function  y=1/x )?

I quickly realized that its volume is actually finite, because to find the volume of revolution the function being integrated has to be squared, which would give 1/x^2, and, as we all know, that converges. So, my question is, are there other functions that share this property? The only family of functions that I know that satisfy this is 1/x, 2/x, 3/x, etc.

Unfortunately, I cannot find a better link for reference than this link, but "Infinite Acres" is a rather old MAA animated short film (part of a larger series, as in the link) about a solid of revolution where the original region has infinite area, the solid has finite volume, and the solid has infinite surface area. – Isaac May 12 '11 at 4:19

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even worse, i once saw this crazy "Infinite Acres" cartoon, it's still available in DVD or VHS, but nobody has been thoughtful enough to stick it on YouTube. this crazy solid has a name:
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I wanted to mention a related topic. You might also note that the surface area of your object is also infinite, despite its finite volume. Thus, if you were to 'hold' such an object, you could fill it with paint but never cover its walls. This has a name - it's Gabriel's Horn ( or Torricelli's Trumpet), and you can read about it here.

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from Wikipedia
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Gabriel's Horn (also called Torricelli's trumpet) is a geometric figure which has infinite surface area, but finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli.

Mathematical definition

Gabriel's horn is formed by taking the graph of

y = 1/x


with the domain x > 1 (thus avoiding the asymptote at x = 0) and rotating it in three dimensions about the x-axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, 
where a > 1 . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume V and the surface area A.