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ROBOT's shortest path in obstacle field mapped by Dijkstra's algorithm. It was long assumed that the most efficient way to find best routes in graphs is Dijkstra algorithm. But it's optimality was proven only in 2023, in a work that won best-paper award. community.wolfram.com/groups/-/m/t/3…
How-to "see" 4th dimension in simple steps: ...using 𝐟𝐚𝐦𝐢𝐥𝐢𝐚𝐫 𝐨𝐛𝐣𝐞𝐜𝐭𝐬. 1️⃣ This 4D object = donut; 2️⃣ Cut typical 3D donut across its tube; 3️⃣ The shape of the cut is usual 2D circle; 4️⃣ Generalize by +1: cut 4D donut and get 3D shapes at the cut. Which is similar…
𝓜𝐚𝐧𝐝𝐞𝐥𝐛𝐫𝐨𝐭 fractal encodes 𝓙𝐮𝐥𝐢𝐚 fractals. Video: 𝐰𝐡𝐢𝐭𝐞 𝐝𝐨𝐭 in 𝓜 defines connectivity of 𝓙. Both 𝓜 and 𝓙 fractals are defined as 𝐙 ↦ 𝐙²+ 𝐂 The difference? 𝓜 plots 𝐂, and 𝓙 plots 𝐙. The white dot (parameter 𝐂) travels in 𝓜 set (corners) and…
𝐓𝐨𝐤𝐲𝐨 𝐬𝐮𝐛𝐰𝐚𝐲 is 99% on-time, world's best. But how lines can be shaped by 𝐄𝐝𝐨 𝐏𝐞𝐫𝐢𝐨𝐝 (1603–1868)? Try to guess before reading further 👇 Each dot is a train (real data) ramping up to one of the world highest peak frequencies. Goes ~40m deep. Tokyo's subway…
🟥 or 🟦 floats above the other? This stereo illusion (𝐜𝐡𝐫𝐨𝐦𝐨𝐬𝐭𝐞𝐫𝐞𝐨𝐬𝐢𝐬) needs both eyes: close one and illusion is gone. 𝐁𝐨𝐢𝐝𝐬 or 𝐬𝐰𝐚𝐫𝐦 𝐢𝐧𝐭𝐞𝐥𝐥𝐢𝐠𝐞𝐧𝐜𝐞 type algorithm used in simulation - expand for tiny code👇 Wolfram Mathematica code (do you…
𝐊𝐞𝐦𝐩𝐞'𝐬 𝐮𝐧𝐢𝐯𝐞𝐫𝐬𝐚𝐥𝐢𝐭𝐲 𝐭𝐡𝐞𝐨𝐫𝐞𝐦: There is a 𝐥𝐢𝐧𝐤𝐚𝐠𝐞 that signs your name. Proves a linkage exists to draw any algebraic planar curve. But to prove is NOT to design elegantly. Novel elegant method: community.wolfram.com/groups/-/m/t/2…
𝐃𝐞𝐥𝐚𝐮𝐧𝐚𝐲 🔲 & 𝐕𝐨𝐫𝐨𝐧𝐨𝐢 🟦 meshes are DUALS of each other. 𝘤𝘰𝘳𝘳𝘦𝘴𝘱𝘰𝘯𝘥𝘪𝘯𝘨 🔲 & 🟦 edges are ⊥. 🔲 vertices <=> 🟦 cells. 🔲 triangles <=> 🟦 vertices. CODE: community.wolfram.com/groups/-/m/t/3…
Twist bottom (left) ring into top shape and its 4 colored lines become 1. Cut along this line - ring remains single object. Cut many times - get 𝐂𝐚𝐧𝐭𝐨𝐫 𝐃𝐮𝐬𝐭 𝐅𝐫𝐚𝐜𝐭𝐚𝐥. 3D objects with 𝐌ö𝐛𝐢𝐮𝐬 𝐬𝐭𝐫𝐢𝐩–𝐥𝐢𝐤𝐞 characteristics. wolfr.am/1us1TUFt1
Haiku for 𝐕𝐨𝐫𝐨𝐧𝐨𝐢 𝐃𝐢𝐚𝐠𝐫𝐚𝐦: 𝘐𝘯𝘬 𝘥𝘳𝘰𝘱𝘴 𝘰𝘧 𝘤𝘰𝘭𝘰𝘳𝘴 • 𝘴𝘱𝘳𝘦𝘢𝘥 𝘵𝘩𝘳𝘰𝘶𝘨𝘩 𝘱𝘢𝘱𝘦𝘳 𝘷𝘦𝘪𝘯𝘴 𝘢𝘯𝘥 𝘮𝘦𝘦𝘵 • 𝘢𝘵 𝘝𝘰𝘳𝘰𝘯𝘰𝘪 𝘸𝘦𝘣 Named after 𝐔𝐤𝐫𝐚𝐢𝐧𝐢𝐚𝐧. Ubiquitous: fire stations, protein packing, more wolfr.am/1ukxO76vZ
🌓TODAY & ONCE in 2.5 years! Grab those grandpa binoculars. Go outside! Bring friends. 𝟏𝟎-𝐬𝐞𝐜𝐨𝐧𝐝 𝐯𝐢𝐝𝐞𝐨 𝐠𝐮𝐢𝐝𝐞, Wolfram-made for 𝘸𝘩𝘢𝘵, 𝘸𝘩𝘦𝘯, 𝘸𝘩𝘦𝘳𝘦 in the sky. Don't miss Americas' night show. Explore: community.wolfram.com/groups/-/m/t/3…
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