Making Fun in Real Life: Tensegrities

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Из курса от партнера University of California San Diego

Mathematical Thinking in Computer Science

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University of California San Diego

Mathematical Thinking in Computer Science

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Курс 1 из 5 — Specialization Introduction to Discrete Mathematics for Computer Science

Mathematical thinking is crucial in all areas of computer science: algorithms, bioinformatics, computer graphics, data science, machine learning, etc. In this course, we will learn the most important tools used in discrete mathematics: induction, recursion, logic, invariants, examples, optimality. We will use these tools to answer typical programming questions like: How can we be certain a solution exists? Am I sure my program computes the optimal answer? Do each of these objects meet the given requirements? In the course, we use a try-this-before-we-explain-everything approach: you will be solving many interactive (and mobile friendly) puzzles that were carefully designed to allow you to invent many of the important ideas and concepts yourself. Prerequisites: 1. We assume only basic math (e.g., we expect you to know what is a square or how to add fractions), common sense and curiosity. 2. Basic programming knowledge is necessary as some quizzes require programming in Python.

Из урока

Making Convincing Arguments

Why some arguments are convincing and some are not? What makes an argument convincing? How to establish your argument in such a way that there is no possible room for doubt left? How mathematical thinking can help with this? In this week we will start digging into these questions. We will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one. Through various examples we will observe a parallel between constructing a rigorous argument and mathematical reasoning.

One Example is Enough3:26

Splitting an Octagon1:24

Making Fun in Real Life: Tensegrities10:25

Know Your Rights5:49

Nobody Can Win All The Time: Nonexisting Examples8:27

Познакомьтесь с преподавателями

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Vladimir Podolskii
Associate Professor
Computer Science Department

Now we switch to some example from the real life.

Sometimes in the real life,

you could be interested in constructing

something which looks strange and impossible at first,

looks impossible at first,

but then you show this and this is funny.

So, this is one of the construction it's called, Tensegrity.

It's a special artificial word,

it's made of tension and integrity.

So, what is Tensegrity?

So, imagine you have a lot of thread and you have some drinking straws,

actually you need only three as it's shown here.

And your goal is to create some solid construction.

Well, these straws do not touch each other but they are just connected by these threads.

You can connect some points there but what you should not do,

you should not do like this.

You cannot tie them together.

You should only make,

they should be on some distance and only you should have the thread.

So, at first it looks impossible but it's not so.

So, now we will try to make a Tensegrity with some things which we have here.

So, the idea is, let me try to explain to you.

We take three things which are somehow like this,

and then the endpoints form something like vertices of an Octahedron.

If they all go for one point it will be an exact Octahedron.

Now, it's kind of approximation.

And the idea is that we connect all of them by our thread.

So, I prepared some kind of needle to make it easier.

But for that we need some things to keep it in the correct positions,

to make holes and so on.

So, this is for making fixed position.

Okay. So, the first thing we should make a hole in the potato to fix them.

Okay, so let me try.

Okay. So, this is our first stage.

Now, they are fixed but now the plan is to connect all the things with a rope.

And for that we need to make holes with a special device, and the candle.

We'll try to make holes.

Actually, I don't know whether it will work now,

because I never tried it seriously in this way, but let's hope.

So,

holes.

So,

it seems we

have all the holes are done.

And actually each vertex will be connected to four others.

So, there are two threads going.

So, we need four holes and we have made them, hopefully.

Okay. So, now we should balance things more or less to make

it symmetric as much as possible.

And then we'll try to make this with a thread.

So, let's see whether it goes through.

Okay. More or less.

Not completely.

Okay. We need a lot of thread.

So,our next stage is just connecting vertices

together, let me

try.

Actually there's a mathematical question.

So, each vertex is connected to four others,

and we need to make,

it is a kind of graph,

and we need to make a path in the graph which with it goes through all the edges.

It's called Eurelian path.

So, and we will know later why it exists,

but now it's enough that it exists.

So,

that's what we achieved,

and it seems more or less okay.

But the question is whether what is holding them

together is this potato side or the ropes.

So, to finally be sure,

we need to kind of cut the potato out.

Which is also a delicate

operation.

So, now we make clear all these things.

But then the question is how we can take the Potato out.

It should somehow get out but not destroying the entire thing.

Okay, like this. Oh, look!

Everything is perfect, you see that there are eight, this is an Octahedron,

there are eight triangle faces

and the big diagonals are made by these straws,

and you see that they don't touch each other,

and the entire thing is rather stable,

you see I can rotate it and move it and everything.

They don't each other anyway.

So, we have a Tensegrity,

and Buckminster Fuller should approve us now I guess. Thanks for watching.

Here's this ready thing.

So you see that there are three straws and they do not touch each other.

Now, it's not, in the video probably it was seen better,

but you see there's some space here, here,

and here, they really don't touch each other.

And if you want to see this cube,

you can look actually a bit different Tensegrity,

you can go to Wikipedia page and there is even animation.

So, compared to our example,

some things missing here.

There is one thread missing here and one missing here.

Topologically, now, if you add these to threads,

topologically the same construction.

But ours is more,

it's an Octahedron which is more regular.

But this is a narrow shape.

Okay. And here is the animation,

you can see how it rotates.

Finally, you can even travel and see a Tensegrity in the real life.

And so, here is the photos,

also from Wikipedia, it's something called Needle Tower.

And this is made by

Kenneth Snelson who was a student of an American architect Buckminster Fuller,

who was the inventor of the word and who made many of things,

many of Tensegrities and placed them in different places.

And this one, you see the two views,

the view from side view, and the view from the bottom.

And it's clear that there are only,

you see these sticks,

now they are serious tubes,

and some threads, or, I don't know what,

ropes, strings or whatever, which connect them.

And as you see this construction is rather stable so it's okay.

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