Whether you want to spice up family photos or give AI images another level of realism, I'll show you two straightforward ways to 3D-ize your images. The result will be something like this: First, pick an image with decent focus and clear boundaries of subjects. Then, you'll use machine-learning/AI tools to generate a depth map to match your image. A depth map is a greyscale image where every pixel encodes the depth of pixels in your original image. In other words, the depth map tool makes ...

As we've seen, just brute-force trying numbers won't find the Collatz unicorn that returns to one. Neither will it prove that such a number CAN'T exist. We need to move beyond calculation using specific numbers and think about and imagine numbers we can only name, numbers without a face. How can that work? How can we talk about numbers (and whole classes of numbers) without naming them exactly? That's the difference between calculation and mathematics. Let's start by looking at the rules of C...

We're already learning a lot about this problem we're up against but if we're going to embark on this journey we need to get a map of the territory. First, there's the mountain range of any given Collatz sequence like 27, as we've seen: [[]] Its parts are: S1 - the start. If we're interested in only rising sequences then S1 must be odd (2X+1) Peak - the highest number reached by a finite Collatz sequence. By definition this must be an even number, otherwise its successor would be higher Drop...

So, let's say we want to disprove Collatz. That means we're in the hunt for an illusive unicorn the first (lowest) starting number that either climbs forever or rises and falls back to itself in a loop. Let's define these kinds of starting numbers: S1 is the starting number of any Collatz sequence (regular finite one that ends in 1 or a unicorn) I1 (Infinite) is the lowest "unicorn" number that climbs forever L1 (Looper) is the lowest "unicorn" number that begins a loop What do we know abou...

I'm using the #100Days challenge to blog my way toward a book for math amateurs exploring the Collatz Conjecture. I'm calling it: Collatz For Crackpots That's a title that requires two definitions: What is Collatz? and Who/What are the Crackpots? First, the Collatz Conjecture has been called the "Easiest, Hardest Math Problem". You can explain it to any precocious 8-year old as follows: Take any number N. If it is even, divide it by 2. If it is odd, multiply by 3 and add 1. Repeat. What ...

American Exceptionalism is like spotty cell phone coverage. At times and places (and for some people more than others), the American "signal" is exceptionally strong. We just need to keep demanding and striving for better coverage. ...