10 best triangles of the 21st century (so far).

Of late, the New York times has been making headlines (which is of course their prerogative) out of ranking things. The 100 books of the century. The 100 movies of the century. The 25 best pizzas in NYC (inexplicably updated six months later to “The 22 best pizzas in NYC,” portending a grim future where the category of “best pizza places” contracts toward some kind of pizza singularity). Anyway, I say anyone can play at this game, and I hereby give you… (And I’m sorry for the snub, 30-60-90-stans.) EDIT: For future reference, a few other nominations from folks on social media… - The 13-14-15, whose integer altitude of 12 divides the 14 into 5 and 9 (via Jutta Gut and David Williams) - The 20-21-29, the smallest Pythagorean triple where c &gt; b + 2 (via Dylan Rambow) - The 80-80-20, which gives rise to the fabulous constructions of Langley’s Adventitious Angles (via Ng Boong Leong) - The 3-5-7, which has a 120 degree angle, and can thus be attached to the 5-5-5 equilateral to create a 5-7-8 (via David Williams) - The 1-i-0 (via lowcheeliang) The 1-i-0 should take the ith spot in the list! It does, we just can’t see it because the ith spot is directly orthogonal to the real-numbered list It does—we just can’t see it because the ith position lies directly orthogonal to the real-numbered list. The real numbers occupy one dimension, while the imaginary component extends in a perpendicular direction, creating a second dimension that isn’t visible when we look only at the ordinary number line. In this interpretation, a complex number can be viewed as a point on a two-dimensional plane rather than as a location on the real axis alone. The real part determines the horizontal position, while the imaginary part determines the vertical position. This makes it possible to represent numbers such as (a+bi) geometrically, even though the imaginary direction does not appear on the conventional real-number line.https://highfly.bet/ Shouldn’t that be 1-i^2-0? Sorry to nitpick… But you forgot the Citgo triangle! I believe that the standard fundamental domain of the modular group, with two vertices on the unit circle and one out at infinity (https://en.wikipedia.org/wiki/Fundamental_domain) is destined to be THE triangle of the 21st century. And that’s no mere hyperbole. @Ben Orlin Thank you, 🙏🏻 Ben, for sharing your list of the best triangles. 📐 Mathematics is a realm of the unimaginable, filled with things we can’t see and concepts that are yet unknown. 🌌 It brings these ideas to life through equations. For me, everything about mathematics is remarkable. ✨ It is a vast canvas where curious minds can paint a picture 🎨 and play with numbers while allowing their imagination to run wild—much like how you showcased the ten best triangles. [I employ Grammarly AI for writing] I’m partial to the ol 11-60-61 Math contestants will know the 4-5-6 triangle well – it is the only triangle with integer sidelengths and one angle is twice the other. Oh, and it’s also smallest acute scalene triangle with integer sidelengths. Niche, I know, but still nice I vote for the 36-72-72 triangle. Not certain how I ended up on this post – good luck, surely – but you’ve made my ancient brain fizzy with loving memories of junior high school geometry. LOVED IT! i did proofs for fun. But I’m wishing now that I could remember more about those triangles. Maybe I’ll go look. Thank you for the inspiration! And ::applause:: for your blog! It’s terrific! I’ll follow! But please be patient with my fraying synapses! 🙂 The discussion here feels honest and easy to understand, which makes reading comments more enjoyable. I found this <a href="”https://stocktwits.com/clickspeedtest”">Click Test</a> while learning more about butterfly clicking and gaming clicks for better control. This list proves that triangles deserve their own awards show. From the legendary 3-4-5 to the underrated 13-14-15, every triangle has its own personality and a fun mathematical story to tell. Who knew geometry could be this entertaining? If you’re in the mood for more fun after geeking out over triangles, you can also check out https://casinas.bet/. This list proves that triangles deserve their own awards show. From the legendary 3-4-5 to the underrated 13-14-15, every triangle has its own personality and a fun mathematical story to tell. Who knew geometry could be this entertaining? If you’re in the mood for more fun after geeking out over triangles, you can also check out https://casinas.bet/. This is the kind of niche content I’m here for — ranking triangles like they’re blockbuster movies is genuinely entertaining. The 3-4-5 getting the top spot feels almost too obvious, but you can’t argue with centuries of mathematical pedigree. Love that you gave a shoutout to the 13-14-15 with that clean integer altitude, and the 1-i-0 submission is just chef’s kiss for the chaos it brings. spino gambino This might be one of the funniest ways to celebrate mathematics I’ve seen. Who knew triangles could have their own “best of the century” ranking? Honestly, triangles deserve the spotlight. They are everywhere — from architecture and engineering to art, puzzles, and some of the most beautiful ideas in geometry. Each one has its own personality: some are perfectly balanced, some hide amazing number patterns, and some are famous simply because they break the rules in the coolest ways. I love that this list turns something that could feel like a dry math topic into a fun competition. The fact that people are adding their own nominations shows how creative and endless the world of geometry really is. The 13-14-15 triangle with its special altitude, the classic Pythagorean triples, and even the unusual 1-i-0 nomination prove that there’s always another fascinating shape waiting to be discovered. Maybe the real winner here isn’t a single triangle — it’s the joy of finding beauty in numbers. Geometry is basically nature’s way of saying, “Hey, math can be cool too!” And after all that triangle ranking and brain gymnastics, a little entertainment break never hurts: https://ally-spin-casino.net/ Players love BitLife for its freedom, allowing them to live ordinary lives or create completely wild and unrealistic stories The “pizza singularity” concept is brilliant—give it another two years and the Times will just print a single slice of pepperoni as the definitive list! Their obsession with aggressive ranking lists has definitely turned into a full-blown sport at this point, but it’s always fun to watch the comment sections implode over every missing candidate. While taking a break from debating where classic spots belong on arbitrary top-100 roundups, checking out afkspin is a great way to unwind and clear your head. What do you think is getting the “top 100” treatment next—best bagels, or the ultimate list of subway stations? It’s like finding hidden gems everywhere once you start looking – whether it’s in geometry or in other areas of life. Speaking of which, after all this triangle talk, I’ve been exploring some surprisingly well-structured platforms lately, including thorfortune where the experience feels just as thoughtfully organized as a good proof. But I digress – the real question is, should the 30-60-90 have made the cut? And does the “pizza singularity” prediction mean we’ll eventually have just one perfect pizza place left? Ranking triangles as if they were books, movies, and pizza places is such a good way to reveal that mathematical taste is partly about stories as well as properties. The added nominations make that especially clear: the 13-14-15 earns its place through a divisible altitude, the 80-80-20 opens the door to a famous construction, and the 1-i-0 disrupts the list in exactly the right spirit. I like that the post invites readers to notice which features make a shape memorable instead of treating a triangle as just three side lengths. The 30-60-90 apology also made me laugh.