{"id":327,"date":"2026-05-29T21:39:20","date_gmt":"2026-05-29T21:39:20","guid":{"rendered":"https:\/\/scienceblog.com\/experimentalfrontiers\/?p=327"},"modified":"2026-05-29T21:39:20","modified_gmt":"2026-05-29T21:39:20","slug":"the-golden-ratio-pythagorean-triples-and-%cf%80-22-7","status":"publish","type":"post","link":"https:\/\/scienceblog.com\/experimentalfrontiers\/2026\/05\/29\/the-golden-ratio-pythagorean-triples-and-%cf%80-22-7\/","title":{"rendered":"The Golden Ratio, Pythagorean Triples, and \u03c0 ~ 22\/7"},"content":{"rendered":"

This is a story of pure math \u2014 not what I usually write about \u2014 but this story captured my imagination yesterday as I was playing with a simple three-line computer program. The result presented a puzzle that requires only high school math to understand, but has no easy answer.<\/strong><\/p>\n

\n
\n<\/div>\n

(I presented the puzzle to Grok this morning, and Grok gave me gibberish as an explanation. Grok retreated when I pointed out that its explanation didn\u2019t hold water.)<\/em><\/p><\/blockquote>\n

Start with the well-known fact that 22\/7 is a good rational estimate for \u03c0.<\/p>\n

22 \/ 7 = 3.142857\u2026
\n\u03c0 = 3.141592\u2026<\/p>\n

You might ask, \u201chow big do the numbers have to get before you have a better estimate than 22\/7? The answer is 179 \/ 57 = 3.140350\u2026<\/p>\n

Generalizing, we might formally define \u201cgood rational estimate\u201d in the natural way \u2014 it\u2019s the ratio of two numbers that gives a closer approximation to \u03c0 than any smaller integers.<\/p>\n

Hold that thought and skip to the Golden Ratio<\/h2>\n
\n
\n
<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

The next step in appreciating the math puzzle is to ask about good rational estimates for the Golden Ratio, \u03c6 = 1.61803\u2026 The answer is<\/p>\n

2 \/ 1 = 2.0
\n3 \/ 2 = 1.5
\n5 \/ 3 = 1.666\u2026
\n8 \/ 5 = 1.6
\n13 \/ 8 = 1.625
\n21 \/ 13 = 1.61538\u2026<\/p>\n

You recognized these numbers immediately as Fibonacci numbers. This is no accident. A common definition of Fibonacci numbers is that each is the sum of the last two. If you see that the ratio of successive Fibonacci numbers is approaching a limit, you can write an equation for what that limit must be:<\/p>\n

\n
\n
\n

\"\"<\/p>\n

<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

where A is any Fibonacci number, B is the next one, and B+A is the next one after that. Now define \u03c6 = B\/A, cross multiply and this equation turns into a quadratic equation for \u03c6. You can solve that equation with the quadratic formula and find<\/p>\n

\n
\n
\n

\"\"<\/p>\n

<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

How closely spaced are the good rational estimates?<\/h2>\n
\n
\n
<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

If I think like a statistician, I would say that getting closer to \u03c6 becomes \u201cexponentially more difficult\u201d in a loose sense of the words. What I mean is that if I already have \u03c6 approximated to 3 digits and I want to do better than that, I\u2019m looking at odds on the order of 1 in 1,000 that the numbers happen to work out. But then if the estimate is good to 4 digits, I face 1\/10,000 odds in finding a better pair of numbers. So, just based on statistics, I would guess that the denominators in the fractions rise approximately exponentially.<\/p>\n

And, of course, we just saw that in the case of the golden ratio \u03c6, the successive good estimate denominators are each about \u03c6 times bigger than the last one, so they are increasing exponentially.<\/p>\n

If we plot the denominators on a log scale, it looks like this. These are the first 35 Fibonacci numbers.<\/p>\n

\n
\n
\n

\"\"<\/p>\n

\n
<\/div>\n<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

The exponential curve comes out a straight line on this log plot. The slope of the line is the log of \u03c6 = 0.481, meaning that each denominator is approximately \u03c6 times bigger than the previous one.<\/p>\n

The statistical argument suggests that the good estimates should increase exponentially for any (irrational) number. We could ask for the first 35 denominators in good estimates of the number e=2.7182845\u2026<\/p>\n

\n
\n
\n

\"\"<\/p>\n

\n
<\/div>\n<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

The line is a little more wobbly, but clearly it is increasing exponentially in the long run. The slope in this case is 0.436, a little smaller than the slope for \u03c6.<\/p>\n

Let\u2019s do the square root of 10, which is quite close to \u03c0.<\/p>\n

\n
\n
\n

\"\"<\/p>\n

\n
<\/div>\n<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

Again, it\u2019s a wobbly line with a constant slope. In this case, the slope is 0.540, a little steeper than the slope for \u03c6.<\/p>\n

I understand generally why the log plots are approximately straight lines, but I don\u2019t understand why the slopes are what they are in each case, and I don\u2019t understand the regularity of the wobbles, or the fact that there is practically no wobble in the plot for \u03c6.<\/p>\n

Now we\u2019re ready for the puzzle<\/h2>\n
\n
\n
<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

Yesterday, the first case I ran was for \u03c0, and the result surprised me. The good approximations are much closer together, especially as you get up past 10,000. In fact, I was only able to do about 35 cases for all the above targets before the numbers got too big for the way the computer language stores integers. But for \u03c0, I could go much further. This is what the log plot looks like:<\/p>\n

\n
\n
\n

\"\"<\/p>\n

\n
<\/div>\n<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

Clearly the numbers are not increasing exponentially. In fact, they seem to be in 5 different regimes, and each regime is approximately quadratic. Here is a plot of the denominators starting at the first 4-digit number. Instead of plotting the log, I have plotted the square root.<\/p>\n

\n
\n
\n

\"\"<\/p>\n

\n
<\/div>\n<\/div>\n<\/div>\n<\/figure>\n<\/div>\n

So, dear reader, why is our statistical expectation met for all the irrational numbers we tried except \u03c0? Why is the density of good estimates for \u03c0 so much higher than for any other irrational number?<\/p>\n

I think I have an idea why this is true. I don\u2019t have ideas about how many more numbers there are that behave like \u03c0, or why the slopes are what they are, or why the plots (except for \u03c6) jog with semi-regular steps.<\/p>\n

I\u2019m interested that Grok did so badly with these questions, and I expect your comments will be much more intelligent.<\/p>\n","protected":false},"excerpt":{"rendered":"

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(I presented the puzzle to … Read more<\/a><\/p>\n","protected":false},"author":65,"featured_media":330,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-327","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-50"],"yoast_head":"\nThe Golden Ratio, Pythagorean Triples, and \u03c0 ~ 22\/7 - Experimental Frontiers, with Josh Mitteldorf<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/scienceblog.com\/experimentalfrontiers\/2026\/05\/29\/the-golden-ratio-pythagorean-triples-and-\u03c0-22-7\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Golden Ratio, Pythagorean Triples, and \u03c0 ~ 22\/7\" \/>\n<meta property=\"og:description\" content=\"This is a story of pure math \u2014 not what I usually write about \u2014 but this story captured my imagination yesterday as I was playing with a simple three-line computer program. 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Much to the surprise of evolutionary biologists, genetic experiments indicate that aging has been selected as an adaptation for its own sake. This poses a conundrum: the impact of aging on individual fitness is wholly negative, so aging must be regarded as a kind of evolutionary altruism. Unlike other forms of evolutionary altruism, aging offers benefits to the community that are weak, and not well focussed on near kin of the altruist. 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The surprising fact that our bodies are genetically programmed to age and to die offers an enormous opportunity for medical intervention. It may be that therapies to slow the progress of aging need not repair or regenerate anything, but only need to interfere with an existing program of self-destruction. Mitteldorf has taught a weekly yoga class for thirty years. He is an advocate for vigorous self care, including exercise, meditation and caloric restriction. After earning a PhD in astrophysicist, Mitteldorf moved to evolutionary biology as a primary field in 1996. He has taught at Harvard, Berkeley, Bryn Mawr, LaSalle and Temple University. He is presently affiliated with MIT as a visiting scholar. In private life, Mitteldorf is an advocate for election integrity as well as public health. He is an avid amateur musician, playing piano in chamber groups, French horn in community orchestras. His two daughters are among the first children adopted from China in the mid-1980s. Much to the surprise of evolutionary biologists, genetic experiments indicate that aging has been selected as an adaptation for its own sake. This poses a conundrum: the impact of aging on individual fitness is wholly negative, so aging must be regarded as a kind of evolutionary altruism. Unlike other forms of evolutionary altruism, aging offers benefits to the community that are weak, and not well focussed on near kin of the altruist. 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