Because humans can't see in more than 3D, it is challenging to make sense of it for the first time. However, there is a simple yet beautiful pattern behind.
This is how the magic is done!
What is a cube in one dimension?
It is simply two vertices connected with a line of unit length.
To move beyond and construct a cube in two dimensions, also known as a square, we simply copy a one-dimensional cube and connect each original vertex with its copy.
(These new edges are colored blue.)
You can probably guess the pattern by now.
Copying a cube's graph and connecting each of its vertices with its corresponding copy brings it to the next dimension.
This is how it looks in 3D.
Repeating this process one more time, we obtain a tesseract, that is, a cube in four dimensions.
(I stretched the new edges a bit to make it easier to see the pattern.)
All 8 of its faces are 3D cubes.
I have always found this pattern quite beautiful.
Geometry intrigued me since I was a child, and when I discovered how to draw a tesseract, I was over the moon.
Small things such as this ignited my desire to be a mathematician, and I still enjoy playing around with fun math.
I regularly post deep dive threads about mathematics and machine learning, explaining (seemingly) complex concepts in a simple way.
If you also love to look beyond the surface and understand how things work, consider giving me a follow!
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"How large that number in the Law of Large Numbers is?"
Sometimes, a thousand samples are large enough. Sometimes, even ten million samples fall short.
How do we know? I'll explain.
First things first: the law of large numbers (LLN).
Roughly speaking, it states that the averages of independent, identically distributed samples converge to the expected value, given that the number of samples grows to infinity.
We are going to dig deeper.
There are two kinds of LLN-s: weak and strong.
The weak law makes a probabilistic statement about the sample averages: it implies that the probability of "the sample average falling farther from the expected value than ε" goes to zero for any ε.
The single biggest argument about statistics: is probability frequentist or Bayesian? It's neither, and I'll explain why.
Buckle up. Deep-dive explanation incoming.
First, let's look at what is probability.
Probability quantitatively measures the likelihood of events, like rolling six with a dice. It's a number between zero and one. This is independent of interpretation; it’s a rule set in stone.
In the language of probability theory, the events are formalized by sets within an event space.
(The event space is also a set, usually denoted by Ω.)