Exploring Mathematics with Matplotlib and Python

((ln n)/(ln ln n))(1 + o(1))

with high probability (i.e., probability 1 - o(1)), where ln denotes the natural logarithm (log to base e). Empirically, the maximum frequency for n=1000, 10000, 100000 often seems to be, respectively, 5, 6 or 7, and 7 or 8.

This problem has applications in studying hash tables etc., and can be found under terms like "maximum load" with balls in bins, and proving this doesn't seem to be very easy. As the post says “the solution is likely not as trivial as it first looks”. The analysis may be hard, but these days if faced with a problem like this in the real world (e.g. we have a hash table of size M that will receive N entries in it, and we're curious about the likely maximum load), we can likely just experiment to find out. Even when the numbers are too large to run simulations directly, an in-between solution is to get a tractable expression (a recurrence relation using dynamic programming or whatever) for the closed form, and write a program to compute it.

Thank you so much for pointing this out! Experimental mathematics feels like a missing puzzle piece in which it makes so much more sense.

Quotes are from the wiki article you linked.

> As expressed by Paul Halmos: "Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."[3]

I wish there were books on how people would describe their complete process (not only their proof) on how they figured things out.

> Mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.

Why is this the case? It seems like it doesn't benefit us other than saving some paper.

> The following mathematicians and computer scientists have made significant contributions to the field of experimental mathematics:

Fabrice Bellard

Donald Knuth

Stephen Wolfram

(among others)

--> This is so awesome, it also sheds some light into how these people think.

Interactive visualisations are often impractical. They don’t work in publications, presentations or documents. Generally speaking, visualisations that have everything clearly visible without requiring interaction are always superior to visualisations that require extra interactions.

I found that it always paid off to do some extra thinking on how to reshape my plots so they don’t need to be interactive. I had very few cases where I needed the plots to be interactive, and ironically, in these occasions only matplotlib worked for me. Those were cases where I wanted to show and play audio snippets that belonged to data points in a dimensionality reduction plot. It was quite hard to get matplotlib to do what I want, but I didn’t even get a anywhere near a result with plotly et al.

There is value is static typing, but there are many instances where that cost is not worth the reward.