More on Threes and Fours

This post is a follow up on my previous post so it will make more sense if that post is read first. I made that post yesterday and today I was surprised to find that the threes and fours were still following me. Today I'm 26204 days old today and this number turns out to be divisible by four. In fact:$$26204=4 \times 6551$$So there we have a very clear connection with the number 4 but what about 3? Well, 26204 has a connection to equilateral triangles because it's a member of OEIS A171971 (see Figure 1).

A171971

Integer part of the area of an equilateral triangle with side length \(n\).

Figure 1

So an equilateral triangle with a side of 246 units has an area of 26204 square units when rounded down to the nearest whole number which closer than 99.999% of the exact value. So that's the connection with the number 3. I know that it could be argued that once you start looking for connections to 3 and 4 in the larger numbers, you'll find them but the connections of 26204 to 3 and 4 are not obscure. The connection with 4 is via its factorisation, the most fundamental characteristic of a number, and the connection to 3 is via its very close approximation to the area of an equilateral triangle with integer sides viz. 246. Of course, I could go further and point out that 246 has three digits and an average digit sum of 4 but that might be overkill.

On this same day, I visited the local McDonalds with my granddaughter. While we there, she did some drawing on her iPad and I continued reading the Pauli and Jung book referred to earlier. I had just finished reading a chapter on synchronicity when we decided to return home. Upon exiting the restaurant, I was struck by another flagrant appearance of 3 and 4 in the carpark. Figure 2 shows the sight that confronted me after descending the front stairs of the building. The signs were right in front of me and stopped me in my tracks. I got my phone out and snapped the photo shown.

Figure 2

As I was writing this post, I was reminded of my own date of birth on the 3rd April 1949 that can be written in dd/mm/yy form as 3/4/49 or mm/dd/yy form as 4/3/49. I also got thinking about my year of birth that is a \(4k+1\) prime number and thus expressible as a sum of two squares. It turns out:$$1949 = 1849+100=43^2+10^2$$Thus 3 and 4 turn up even my year of birth. I seem to be on a 3-4 roll at the moment.

The tetrahedron seems to be the perfect fusion of the 3 and 4 numbers, having four triangular faces. See Figure 3 and Figure 4.

Figure 3: source

Figure 4: tetrahedron net (source)