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  • Whoa!

  • Help me!

  • I'm being mugged!

  • Travel mugged, that is!

  • This is the Twizz mug by neolid.

  • This video is not sponsored by the way.

  • I bought this mug myself after seeing someone talk about it on Twitter because I was intrigued

  • by the way it closes.

  • We're going to talk about the math behind the way it seals today on Michael's toys.

  • As you can see when it's closed it looks a little bit likeUranus has a diameter

  • of more than 50,000 kilometers.

  • That's more than a mile.

  • That's a pretty out of this world fact but the way the Twizz mug closes is very much

  • of this world.

  • It may look strange but we use this kind of seal quite often.

  • We do every time we close up a bag of bread.

  • When we do that we're taking a sleeve of material and we're giving it a twist.

  • This seal is pretty nice but from above it can look a bit like a Chocolate starfish and

  • the hot dog-flavored water was the title of Limp Biscuit's third studio album.

  • It came out in the year 2000 and it was extremely popular with middle schoolers named Michael

  • Stevens who lived in Stilwell,Kansas.

  • But we are here today to talk about this mug.

  • Let's go back to the bread bag first though.

  • When I close up a bread bag I am taking essentially as I said a cylinder of flexible material

  • and I'm rotating the top such that I get a nice tight seal down here but notice that

  • when I do that rotation the height of the bag's material shrinks.

  • It begins this tall and it ends up shorter.

  • Lengths of material that originally spent all that length going up, when twisted have

  • to spend some of that length going forward and right and backwards and left and they

  • have less of their length to go straight up so the bag becomes shorter.

  • If I need extra material to keep the height where I want it why don't I just cuff the

  • material.

  • If I form a little cuff like that then as I close the bag and it gets shorter I can

  • surrender some of the material from the cuff and make the cuff shorter and raise it back

  • up to the height that I want.

  • That's exactly what happens in this mug.

  • We have a cylinder of rubbery material, the bottom of which, like the bottom of this bag

  • analogously is connected inside the mug.

  • The top, the top folds over the lip of the mug and forms a cuff where it then attaches

  • to this external ring so when I twist the material, the cylinder of material, the external

  • ring goes up surrendering material to the inside and we get ourselves a nice seal.

  • Now the mathematical name for the surface created when you rotate one end of the cylinder

  • is a hyperboloid.

  • Wow that really does look like a leather cheerio.

  • Now I've seen everything.

  • There are a lot of different ways to generate a hyperboloid.

  • You can take a cylinder and rotate one of the bases.

  • If you rotate a base of a cylinder pi radians every line will intersect at a single point.

  • That's a nice seal.

  • On the Twizz mug if you rotate pi radians which is 180 degrees half of a full rotation

  • it closes up.

  • But the mug itself is actually designed to rotate a bit more than that so you get a really

  • nice seal that is incredibly water tight.

  • It's quite impressive but that's not the way of generating a hyperboloid that gives

  • the hyperboloid its name.

  • To figure out where its name comes from we should look at conic sections.

  • Imagine a double cone, two coins joined tip to tip with parallel bases.

  • Let us now intersect this double cone with a plane.

  • If the plane is parallel to the bases the intersection will be a circle or in one case,

  • a point.

  • If the plane however is not parallel to the bases but its slope is less than the slope

  • of the cones the intersection will be an ellipse.

  • If the slope of the plane is equal to the slope of the cones the intersection will either

  • be a line or in every other case a parabola.

  • Parabola means to throw next to.

  • To throw near.

  • It's the shape of the trajectory of an object thrown here on Earth's surface.

  • If however the plane has a slope that is greater than the slope of the cones it will intersect

  • both cones and the intersection is called a hyperbola.

  • Where hyperbola means to throw beyond.

  • We get the word hyperbole from the same roots.

  • It means to have kind of gone too far.

  • To have exaggerated.

  • To have been hyperbolic.

  • And watch this.

  • If we take a hyperbola and rotate it around this axis we get a hyperboloid of two sheets

  • but if we rotate the hyperbola around this axis we get a hyperboloid of just one sheet

  • and this is what our mug is making.

  • The oid ending that we've ended to hyperbola simply means akin to.

  • Kind of looks like.

  • So a hyperboloid kinda looks like a hyperbola but like it's not because it's a surface

  • not a one dimensional thing.

  • Any who a hyperboloid of one sheet is a ruled surface which means that it can be constructed

  • by taking a straight line and moving that straight line through space.

  • This is very important because if you need to build a structure that curves it's pretty

  • economical to use the hyperboloid because you could just make it out of straight materials.

  • You don't have to build special curved materials.

  • Cooling towers want big wide bases so that there's a lot of surface area for evaporation

  • but it's awesome if they can be a little bit cinched in the middle because as all the

  • stuff evaporates it'll rise up and speed up as it reaches this narrower region due

  • to the Venturi Effect.

  • Then the cooling tower expands which increases surface area again allowing for more cooling

  • giving us condensation and evaporation of all the things that we want the cooling tower

  • to do.

  • And all of it can be made with straight beams.

  • One of my favorite ways to exploit the construction of a hyperboloid is the hyperbolic slot.

  • A straight line magically seems to fit through a curved slot.

  • This of course is merely a demonstration of the fact that a hyperboloid of one sheet is

  • a ruled surface.

  • One of my favorite applications of a hyperbolic slot was used in a display by Hermes where

  • an umbrella passes through a slot shaped like a hyperbola.

  • Instructables has instructions on how to make your own out of leggo.

  • Oh and by the way two axels that are skewed to one another can still contact one another

  • if their gears are hyperbolic.

  • So hyperboloids are all around us.

  • They're beautiful and very useful for when you want a container that can hold liquid,

  • allow you to pour the liquid and when you don't the liquid to spill out seal up.

  • So thank you hyperboloid.

  • I think you're beautiful even when one of your bases is rotated pi radians or more from

  • above you look a little bit like a be whole, be complete, be 100% and as always thanks

  • for watching.

Whoa!

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ハイパーボロイド (Hyperboloids)

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    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語