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Hyperboloids

Hyperboloids


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 like…Uranus 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.

40 comments

when i noticed that it wasnt actually a still frame when "michaels toys" was written on the screen, but the video was still going and i saw his face i diedd. love this guy

Man, gotta tell you. YOU MADE ME LAUGH SO LOUD with these cuts. XDDDD thanks for this. And… oh, yeah, also for the videos as well 😀

Michal I think you’ve been a meme for so long that your subconscious is trying to act like that because it feels it’s supposed to

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