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. https://t.co/0JxgAvqRBW

⚛️ In 𝐫𝐢𝐯𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 of 𝐛𝐥𝐚𝐜𝐤 𝐡𝐨𝐥𝐞𝐬 space itself flows... River of space falls into black hole at Newtonian escape velocity, hitting light speed at horizon. Newton particle-grid with Wolfram differential equations gives a qualitative proxy for the visual: 🔴

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: https://t.co/VsT4Xw1SDM

𝐖𝐨𝐥𝐟𝐫𝐚𝐦 code is very short: 𝕒[1] = RandomReal[1, 20 {1, 1, 1}]; 𝕒[n_] := 𝕒[n] = Rescale[𝕒[n - 1] - GradientFilter[𝕒[n - 1], 2]] Manipulate[𝕒[k]^10 // Image3D, {k, 1, 700, 1}] Full original discussion by 𝐒𝐢𝐦𝐨𝐧 𝐖𝐨𝐨𝐝𝐬: https://t.co/ob7phnKluR

In the video this recursion is applied many times to a 3D table of values, shown as 3D image. 𝐒𝐭𝐚𝐛𝐥𝐞 𝐬𝐩𝐢𝐫𝐚𝐥𝐬 𝐞𝐦𝐞𝐫𝐠𝐞. Video runs evolution backwards to the initial random values, and then forward in time with slightly different visualization technique.

𝐌𝐚𝐭𝐡 𝐃𝐞𝐭𝐞𝐜𝐭𝐢𝐯𝐞 wanted! Why recursion makes spirals? a[n_] := Rescale[ a[n - 1] - GradientFilter[a[n - 1], 2] ] 1️⃣ 𝐚 --matrix or tensor of values ∈ [0, 1] 2️⃣ 𝐑𝐞𝐬𝐜𝐚𝐥𝐞 --keeps 𝐚 values ∈ [0, 1] 3️⃣ 𝐆𝐫𝐚𝐝𝐢𝐞𝐧𝐭𝐅𝐢𝐥𝐭𝐞𝐫 --discrete data gradient

𝐃𝐞𝐥𝐚𝐮𝐧𝐚𝐲 🔲 & 𝐕𝐨𝐫𝐨𝐧𝐨𝐢 🟦 meshes are DUALS of each other. 𝘤𝘰𝘳𝘳𝘦𝘴𝘱𝘰𝘯𝘥𝘪𝘯𝘨 🔲 & 🟦 edges are ⊥. 🔲 vertices <=> 🟦 cells. 🔲 triangles <=> 🟦 vertices. CODE: https://t.co/rpJFboewkQ

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. https://t.co/dRlSDyDQ8F

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 https://t.co/y9Jx2NbJco

🌓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: https://t.co/k5VtON3uaV

harmful proteins. Built in Wolfram Language with GPU acceleration, this method accelerates simulation of rare transitions. Valuable for drug design. On image attached: putative low free-energy path of the CRBN open-close conformational transition. https://t.co/D7N51vj7UB

landscape for physically realistic protein changes. Cereblon protein (CRBN) plays important role in targeted protein degradation. The research maps its transition between open and closed states, a process critical for how molecular glue drugs trigger the breakdown of...

Tech details... This work boosts (classical) molecular dynamics with AI using a physics-informed autoencoder neural network that compresses complex 3D protein structures into a simplified latent space. Key idea is aligning the latent space with the protein’s free energy...

AlphaFold2 (2024 Nobel Prize) predicts protein shape. This AI predicts motion—how proteins shift between shapes. Motion is key to drug design. Wolfram neural net accelerates the search for optimal transitions. https://t.co/D7N51vjFK9

...and here for the context “All roads lead to Lyon” Great examples of "large city" road cluster. In contrast my map is a cluster around population zero. https://t.co/2f93FQsJCD

For the context this is “All roads lead to Rome”: https://t.co/GTpyjSAoAI

In my article, I compute shortest paths from over 37,000 North American cities to Nothing, Arizona—a ghost town with zero population—to show that the clustering effect is independent of the destination’s size, offset maybe only nearest highway intersection.