Reality is what it is

From Peter G. Casazza’s occasionally pessimistic A Mathematician’s Survival Guide (last emphasis mine):

Wonderful Advances. When I first joined the mathematics community I was excited to join a group dedicated to advancing mathematics. I had a rude awakening when it became clear that we were really working to advance ourselves. This is an unfortunate consequence of the reality around us. We must all compete for very scarce research grants, positions, promotion, tenure, awards, raises etc. But we need to be careful that this reality does not diminish our enjoyment of the subject.

One of the things I wish I had read a few years ago. Little niggles can also matter.

How does Tupper’s self-referential formula work?

[I write this post with a certain degree of embarrassment, because in the end it turns out (1) to be more simple than I anticipated, and (2) already done before, as I could have found if I had internet access when I did this. :-)]

The so-called “Tupper’s self-referential formula” is the following, due to Jeff Tupper.

Graph the set of all points such that

in the region

where N is the following 544-digit integer:
48584506361897134235820959624942020445814005879832445494830930850619
34704708809928450644769865524364849997247024915119110411605739177407
85691975432657185544205721044573588368182982375413963433822519945219
16512843483329051311931999535024137587652392648746133949068701305622
95813219481113685339535565290850023875092856892694555974281546386510
73004910672305893358605254409666435126534936364395712556569593681518
43348576052669401612512669514215505395545191537854575257565907405401
57929001765967965480064427829131488548259914721248506352686630476300

The result is the following graph:

Figure 1: The graph of the formula, in some obscure region, is a picture of the formula itself.

Whoa. How does this work?

At first sight this is rather too incredible for words.

But after a few moments we can begin to guess what is going on, and see that—while clever—this is perhaps not so extraordinary after all. So let us calmly try to reverse-engineer this feat.