| Here's an intuitive description of the entropy, [log(log(n)) -sum(log(p_i) log(log(p_i)))]: 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)
(Visualization not exact due to rounding of logarithms, and because)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. |