Actual fact: California’s New Bill Requires that 3D Printers Get DOJ Approval as Firearm-Blocking"

(The "report on themselves" is fiction invented by Adafruit.)

That's not what the (clumsily written) article is about. It's not about sampling the integers to choose an integer, it's about sampling the prime factors of an integer, as a measure of evenness of distribution of prime factors.

It's measuring the information in the prime factorization, the information in the number as a value.

The entropy of a random integer N is the volume of the gap between how much space N takes up and how much space its internal components take up.

This can be visualized as The City of N, in base 2. (OP used log_e, but that's too hard to draw.)

1. The Foundation (The Factors)

Take a random number N and break it into its prime factors. We write these prime factors in binary, side-by-side, along the bottom of a page.

The total width of this baseline is roughly log_2(N)(the number of bits in N).

2. The Cloud Ceiling (The Potential)

We write down the length of (N written down in base 2) in base 2. (If N = 46 = 101110 (base 2), its length is ~6 = 110 (base 2),

We write that number vertically (110) to set the Maximum Ceiling Height.

Finally, we look at the number N itself.

3. The Buildings (The Structure):

Above each prime factor, we construct a building.

* The Width: The width of the building is simply the length of that prime factor in bits.

* The Height: To determine how tall the building is, we look at its width and write that number down vertically in binary.

To normalize, we zoom our camera so the length (log) of N fills the view.

The Entropy (The Visible Sky):

The Sky: This is the empty space between the tops of the buildings and the top of the picture (cloud ceiling).

The Entropy of N is exactly the total area of the visible sky.

If N is prime, the building is as wide and tall as the whole city and touches the cloud ceiling. No Sky. Zero Entropy.

If N is a random integer, it usually has one wide building (the largest prime) that is almost as tall as the ceiling, and a few tiny huts (small primes) that leave a massive gap of blue sky above them.

Here is the visualization for N = 46. (Binary 101110, length ~6).

          (46)
      1 0 1 1 1 0  (length ~5.5)
     ---------------------------
         |1 1 1 1 1|
         |0 0 0 0 0|  
     |1 1|1 1 1 1 1|
     +---+---------+
     |1 0|1 0 1 1 1|
      (2)    (23)

Interpretation:

Building 23 is tall. It reaches Level 3 (101 is length 5). It touches the ceiling (Level 3). There is zero sky above it.

Building 2 is short. It only reaches Level 2 (10 is length 2). There is one unit of sky visible above it.

Total Entropy: The total empty area above the buildings is small (just that gap above factor 2), which matches the math: 46 is "low entropy" because it is dominated by the large factor 23.

A number with High Entropy would look like a row of low, equal-height huts, leaving a massive amount of open sky above the entire city.

A much better way to analyze it geometrically. The 6D problem has 2D trivial symmetries, and is parameterized by 4 polar coordinates: circumcenter distance from origin, circumradius, and 2 internal angles of the triangle. Then the solution is just [expected value of the area of a triangle on unit circle] times the radius integral in the OP article, divided by the π³ volume of the triangle space in rectangular coordinates.

This is totally disqualifying:

> Frankly, I would have used my body to stop the bullet on its path to kill a young father and husband like Charlie at age 31.

No one can jump in front of a single sniper bullet.

Thousands of young fathers die every day. Why isn't Loeb helping them instead of spending all this time promoting his search for aliens, and misleading journalists and the public about extraterrestrial intelligence?

http://neal.fun/infinite-craft

This is what can happen when you live on a campus and don’t waste your life commuting.

It is standard in empirical science to focus on first order effects, and then to address 2nd-order effects, which the book does in the following pages.

Page 10 is a talking about how the angle of shadow is independent of subject position (human or pole), on the order of ~10m, and so we can treat all shadows at the same moment in time in the local area as similar. (Obviously shadows are not similar when comparing someone in Russia to someone in Mexico at the same time.) This is in contrast to a shadow from a streetlamp, where the angle changes from 45 degrees to 80 degrees over the span of a few meters of relative subject position.

This is true, and the 0.5-degree error due to the size of the sun is not important to the overall result, which is already appromixate to a similar level of precision, and is the same for all subjects.