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https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1670129126585-T4BK972QP831VM43UYH4/210929_22+%280-00-17-07%29+copy.jpg art_page_gallery https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1670129188178-AINZ09CW44NLULEDOHPL/Screen+Shot+2022-11-30+at+1.36.33+AM+copy.jpg art_page_gallery https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1670130898133-5LXI4LLIP7ARK45P9VB2/saveacopy.jpg art_page_gallery https://www.billnoll.com/tonic_sample_gallery daily 0.75 2025-03-06 https://www.billnoll.com/sample daily 0.75 2025-03-06 https://www.billnoll.com/visuals daily 0.75 2025-12-09 https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/2a08b75e-2edf-4f9d-b69d-366b7a0d2e5e/site_image.jpg visuals Cross-Section of a Hypersphere A higher-dimensional object can only be observed in our space as a sequence of slices. A 3-sphere, the boundary of a four-dimensional ball, would pass through three-dimensional space as a point that grows into a sphere and contracts again; this ring is one such cross-section. Whether the universe itself is a finite 3-sphere remains an open question. Current measurements favor flat geometry, but searches for repeating patterns in the cosmic microwave background continue. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/713cbed7-2447-4892-a186-ccdc8fd4dffd/190816_71+%280-00-10-08%29.jpg visuals Negative Curvature Each fold here is a saddle, a surface of negative Gaussian curvature that bends up along one axis and down along the other. Such surfaces are the local model of hyperbolic geometry, where parallel lines diverge and area grows exponentially with radius. Nature uses them in coral, kale and sea slugs to pack surface into limited volume. Physicists have recently built hyperbolic lattices in superconducting circuits, opening an experimental window onto geometries no laboratory could previously reach. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/20f48894-5139-47a0-ba8c-e29b0b1ebfe9/Screen+Shot+2024-07-02+at+3.53.01+PM.jpg visuals Epicycles Each rosette is traced by circles moving on circles, a construction known to Ptolemy and later generalized by Fourier: any periodic motion can be decomposed into a sum of rotations. The same principle underlies audio compression, MRI and the analysis of starlight. Viewed as a field, the rosettes resemble the bubble universes predicted by eternal inflation. That theory remains untested, though cosmologists are searching the microwave background for the faint scar a neighboring bubble collision might leave. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/bbcc409e-77b8-4a33-8d3e-869ae0d73c9f/240614_20+%280-00-06-24%29_1.jpg visuals Frame Dragging General relativity predicts that a rotating mass twists the spacetime around it, the Lense-Thirring effect. Gravity Probe B confirmed it around Earth in 2011, where the effect is minute. Near a spinning black hole it dominates: within the ergosphere nothing can remain stationary. The next generation of event-horizon imaging and the LISA gravitational-wave observatory aim to measure black hole spin directly, testing whether these objects are exactly as Einstein's equations describe. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/ba298510-6853-4b3f-8416-5bf6b10f29bd/Screen+Shot+2024-07-02+at+3.40.28+PM.jpg visuals Wavefunction The banding resembles an interference pattern, the signature of a quantum wavefunction. A point often overlooked is that the wavefunction of N particles does not live in ordinary space but in a configuration space of 3N dimensions. Whether that high-dimensional object is physically real, or only a bookkeeping device, divides physicists and philosophers. Experiments testing objective collapse models may eventually decide whether the wavefunction ever truly collapses, or whether every outcome persists. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/09cc69b8-c218-4aca-9e38-5a80725d5bdb/250920_31.00_00_20_11.Still003.jpg visuals Envelope of a Rotating Tesseract Every line in this image is straight. The circle at the center is an envelope, the curve tangent to an entire family of lines, a construction studied since the seventeenth century. The rotating squares suggest the projection of a tesseract, the four-dimensional cube, turning through our space. If additional spatial dimensions exist, gravity should deviate from the inverse-square law at short distances. Torsion-balance experiments have tested this down to tens of micrometers, and the search continues below that scale. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/ae2c6a4c-0aaa-4d2b-9db4-7277950b0813/Screen+Shot+2024-07-02+at+4.17.53+PM.jpg visuals Particle Tracks The streaks recall bubble chamber photographs, in which charged particles left trails of vapor through superheated liquid. Physicists inferred mass, charge and lifetime from the curvature of each track, and in 1964 discovered the omega-minus baryon this way, confirming the quark model's predictions. The method's descendants are liquid-argon detectors, such as the DUNE experiment now under construction, designed to determine whether neutrinos explain why the universe contains matter rather than nothing. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/243c2679-3d78-4554-9719-d469acd81f2b/251204_22.00_00_15_03.Still002.jpg visuals Quasicrystal Seed Fivefold symmetry cannot tile a periodic crystal, yet Dan Shechtman observed it in an alloy in 1982 and received the Nobel Prize in 2011. The explanation is dimensional: a quasicrystal is a slice through a periodic lattice in a higher-dimensional space, projected into our three. Natural quasicrystals have since been found in a meteorite. Their unusual hardness and low friction are now being studied for coatings and photonics, with many structures still unexplored. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/276cd991-a5f3-4c1b-a68b-e7541d4c0b65/Screen+Shot+2024-07-02+at+3.54.10+PM.jpg visuals Dihedral Symmetry This pattern is invariant under eight rotations and eight reflections, together forming the dihedral group of order 16. Group theory turns such symmetries into algebra, and it is the language of modern physics: the Standard Model is built on the gauge group SU(3)×SU(2)×U(1). Grand unified theories propose a single larger group behind all three. If correct, protons should eventually decay, a prediction Japan's Hyper-Kamiokande detector will test with unprecedented sensitivity. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/2411a26d-9c46-4583-97f6-772e1aef334c/Screen+Shot+2024-07-02+at+6.13.00+PM.jpg visuals Einstein-Rosen Bridge In 1935 Einstein and Rosen showed that the equations of general relativity permit a bridge connecting two regions of spacetime. Such a bridge cannot be traversed; it pinches off faster than light could cross it. The idea has returned in an unexpected form. The ER=EPR conjecture, proposed in 2013, suggests that every pair of entangled particles is joined by a microscopic bridge of this kind, meaning spacetime geometry itself may be built from quantum entanglement. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/249119e9-d4d2-49d2-bbef-784dea75995c/Screen+Shot+2024-07-02+at+3.55.41+PM.jpg visuals Hopf Fibration In 1931 Heinz Hopf showed that the 3-sphere, the boundary of a four-dimensional ball, can be filled entirely by circles, any two of which are linked exactly once. The structure appears throughout physics: in the states of a spin, in rigid-body rotation, and in solutions of Maxwell's equations where every field line forms a closed, linked loop. Knotted field configurations of this type, called hopfions, have recently been observed in magnetic materials and are candidates for future data storage. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/22fd177c-3141-42d5-9375-837069a64b81/Screen+Shot+2024-07-02+at+3.51.40+PM.jpg visuals Wallpaper Group There are exactly seventeen distinct ways to fill a plane with a repeating symmetric pattern, a classification completed by Fedorov in 1891. In three dimensions there are 230 crystallographic space groups, and in four dimensions nearly five thousand. Physicists now engineer synthetic dimensions in photonic and atomic systems, allowing them to study the physics of four-dimensional crystals in the laboratory. Which of those symmetries produce genuinely new states of matter is still being mapped. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/c0c05386-ce4a-4e8f-87bd-0b49226dc694/Screen+Shot+2024-07-02+at+3.37.23+PM.jpg visuals Quantized Shells The concentric rings recall the central result of early quantum theory: an electron bound to a nucleus can occupy only discrete energy levels, fixed by the standing-wave solutions of the Schrödinger equation. Hydrogen's spectrum is now measured to roughly fifteen significant figures, making it one of the most precise tests of physical theory. Small discrepancies in those measurements, such as the recent proton radius puzzle, are where new physics would first become visible. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/a911f418-595d-42b7-baba-930594815a3b/Screen+Shot+2024-07-02+at+3.24.11+PM.jpg visuals Latent Space A neural network represents what it learns as points in a space of hundreds or thousands of dimensions. The manifold hypothesis holds that meaningful data occupies only thin, curved surfaces within that space, much as this field of points gathers into orbits around an empty center. Why such representations generalize so well is not yet understood. Mapping their geometry is one of the central problems in machine learning, and may also inform how biological brains encode information. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/d5d6914a-af0c-4e0d-95fb-01eb4adcd3e7/Screen+Shot+2024-07-02+at+3.35.36+PM.jpg visuals Branching Network Branching structures recur across scales because they solve the same optimization problem: reaching the most volume with the least material. Neurons, blood vessels and river systems all follow similar scaling laws. A 2020 quantitative comparison found that the large-scale cosmic web of galaxies and the neuronal network of the brain share comparable statistical structure. Whether that resemblance reflects a deeper principle of self-organization or a coincidence of scale is an open question. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/4a89d63d-d774-4c8e-a8e0-4e4c743cc3e6/Screen+Shot+2025-11-13+at+1.31.58+PM+1.jpg visuals Resonant Mode A vibrating membrane settles into standing-wave patterns called normal modes, which Ernst Chladni made visible with sand in 1787. In 1966 Mark Kac asked whether one can determine a drum's shape from its frequencies alone; in 1992 the answer was shown to be no. String theory proposes that every elementary particle is a vibrational mode of an extended object. Its predictions remain beyond experimental reach, and finding a measurable consequence is one of the field's principal goals. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/4900bfaf-eab8-49a3-9e88-5d0a8935eea0/Screen+Shot+2024-07-02+at+3.50.15+PM.jpg visuals Superposition A quantum system can exist in a superposition of states until measured, a principle confirmed in every regime tested so far. Interference has been demonstrated with molecules of more than two thousand atoms. The open question is whether there is a limit. Proposed experiments aim to place small masses in superposition and observe whether gravity can entangle them, which would provide the first direct evidence that gravity itself obeys quantum mechanics. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/63e3785d-665b-46ab-8082-d9cbb5b8f11b/Screen+Shot+2024-07-03+at+4.34.39+PM.jpg visuals Double Rotation In three dimensions every rotation has an axis. In four dimensions this is no longer true: a general rotation turns two perpendicular planes independently, at two different rates, leaving no point fixed except the center. These double rotations are described naturally by quaternions, discovered by Hamilton in 1843. The same mathematics governs electron spin, which returns to its original state only after two full turns, and it now underlies the control of qubits in quantum computers. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/443989af-2a78-4818-8921-a8f13c9cc120/Screen+Shot+2024-07-03+at+2.06.04+PM.jpg visuals Klein Bottle The Klein bottle is a closed surface with no inside or outside, described by Felix Klein in 1882. It cannot exist in three dimensions without passing through itself, but sits cleanly in four. Such non-orientable surfaces were long regarded as pure abstractions. Recent work in condensed-matter physics has found that the space of electron momenta in certain engineered materials takes exactly this shape, producing electronic behavior that ordinary topology does not allow. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/16ab1ac9-5bcc-4cef-b03e-ccf28a86b6b7/Screen+Shot+2024-07-03+at+2.05.18+PM.jpg visuals Einstein Ring When a massive galaxy lies directly between us and a more distant source, its gravity bends the light into an arc or, with exact alignment, a complete ring. Einstein predicted the effect in 1936 but doubted it would ever be observed; the first full ring was imaged in 1988. Gravitational lensing is now a primary tool for mapping dark matter. The Euclid space telescope is expected to find on the order of 100,000 new lenses, enough to test what dark matter is. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/41519df3-d2d5-4186-8a55-f989de93ba8d/Screen+Shot+2024-07-02+at+3.54.51+PM.jpg visuals Projection of a Hypercube A four-dimensional hypercube cannot be seen directly; we observe only its shadow, a cube within a cube that deforms as the object rotates. Plato's allegory of the cave made the same point about perception two millennia earlier. Modern physics has inverted the idea. The holographic principle proposes that the contents of a region of space are fully encoded on its lower-dimensional boundary, so our three-dimensional world may itself be the projection. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1cb2f628-e025-4a76-a1e5-52464f81ccd5/fjusurface_p6mTruncatedTriHexagon_tiling_2500x1406.png visuals https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/de047fd0-7afe-428d-8711-28d6523559d4/Screen+Shot+2024-07-03+at+2.11.42+PM.jpg visuals Caustic When light is focused by a curved surface, it concentrates along bright curves called caustics, the lines seen at the bottom of a swimming pool. René Thom's catastrophe theory showed that only a small number of stable caustic forms exist, the fold and the cusp being the simplest. Caustics also occur in gravitational lensing, where they can magnify a single star billions of times. In 2022 this allowed astronomers to observe Earendel, the most distant individual star yet detected. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/6a0775ab-aec7-41d2-b381-c7a5046d884c/Screen+Shot+2024-07-03+at+2.09.08+PM.jpg visuals Triangulated Sphere Any surface can be divided into triangles, and for every triangulation of a sphere the number of vertices minus edges plus faces equals two. Euler's formula was the first topological invariant: a quantity unchanged by any continuous deformation. Its modern successor, topological data analysis, counts holes and loops in high-dimensional data. Applied to recordings of brain activity, it has begun to reveal geometric structure in neural networks that conventional statistics do not detect. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/f2b52cf2-b9ed-48d8-8865-511e0928625c/Screen+Shot+2024-07-03+at+2.19.56+PM.jpg visuals Bloch Sphere The state of a single qubit can be represented as a point on a sphere: the poles correspond to 0 and 1, and every other point is a superposition of the two. Quantum computation proceeds by rotating that point. The principal obstacle is noise, which drifts the state away from where it should be. Building error-corrected machines with thousands of reliable qubits is the defining engineering challenge of the field and would make problems in chemistry and materials tractable for the first time. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/0f79ccfc-7534-40fa-8854-59ab9d990e5d/Screen+Shot+2024-07-03+at+2.15.09+PM.jpg visuals Quantum Foam In 1955 John Wheeler proposed that at the Planck length, about 10^-35 meters, spacetime ceases to be smooth and fluctuates violently. This scale is far beyond any accelerator. It may still be testable indirectly: if spacetime is granular, photons of different energies could travel at slightly different speeds over cosmological distances. Observations of distant gamma-ray bursts have already tightened those limits, and future measurements may detect the first signature of quantum gravity. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/84287f28-4cdd-43aa-8ec7-53a778ac60e5/surface_p31m_tiling_2500x1406.png visuals https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/92f90c1a-f14c-4dac-9c52-3f31f47cc5cc/surface_p6m_tiling_2500x1406.png visuals https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/12cad31d-590e-49f4-9e3b-462329e8ba62/251027_52+%280-00-04-06%29.jpg visuals Logarithmic Spiral The logarithmic spiral is the only curve that remains geometrically similar to itself at every scale. Jacob Bernoulli, who called it spira mirabilis, had it engraved on his tombstone. It describes the growth of shells, the paths of storms and, approximately, the arms of spiral galaxies. Why galaxies form and sustain spiral structure is still debated, and the James Webb Space Telescope is finding well-formed disks far earlier in cosmic history than models predicted. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/deed763b-f5de-4d12-9188-da7b709c4266/Screen+Shot+2024-07-03+at+2.20.47+PM.jpg visuals Archimedes' Polygons Around 250 BC Archimedes bounded pi by inscribing and circumscribing polygons of up to 96 sides around a circle, proving that it lies between 3 10/71 and 3 1/7. The method anticipated the limits of calculus by nearly two thousand years. Pi has since been computed to more than a hundred trillion digits, yet a basic question remains unanswered: whether every finite sequence of digits appears in it equally often. No one has proved that pi is normal. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/dab457c6-eaec-4df0-9951-bc0861aea7b5/surface_p6mTruncatedTriHexagon_tiling_2500x1406.png visuals https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/67281c28-c93a-4078-961c-05a9dae67b0c/Screen+Shot+2024-07-02+at+3.43.07+PM.jpg visuals Gradient Descent Neural networks learn by gradient descent, repeatedly adjusting millions of parameters in the direction that most reduces error. Intuition from three dimensions suggests the process should become trapped in local valleys. In very high dimensions it rarely does: most critical points turn out to be saddles, which offer an escape route. Why the solutions found this way generalize to new data, rather than merely memorizing the old, is one of the major unsolved problems in the theory of learning. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/980fe1f7-c92e-4205-ad3d-2a5d4c5e0106/250920_31.00_00_03_14.Still001.jpg visuals Möbius Surface The form is a ruled surface, curved yet built entirely from straight lines, twisted in the manner of a Möbius strip: a surface with one side and one edge, described by August Möbius in 1858. Chemists synthesized the first stable Möbius-shaped molecule in 2003, and the twist alters its electronic properties. Similar topology now appears in proposed photonic and electronic materials, where a single half-twist may control how energy flows. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/22ad8917-dbae-40b1-bf6f-13d3f3e72a7d/Screen+Shot+2024-07-03+at+2.17.50+PM.jpg visuals Hidden Layers Glowing nodes hang inside dotted rings, linked by faint tubes of light in a loose, many-colored cluster. A neural network is built in layers of nodes, and the ones between input and output, where the real thinking happens, are called hidden layers. No one can quite see what they've learned. This is an imagined portrait of one, lit from within by its own connections. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/df6dfb2b-8492-4cb0-b057-9d681261adb4/Screen+Shot+2025-11-13+at+1.33.38+PM.jpg visuals Strange Attractor Golden loops wind around a pinwheel center, crossing and recrossing but never retracing the same path. In chaos theory, a strange attractor is a shape that a system orbits forever without repeating, unpredictable yet never wandering away. Weather, heartbeats and dripping faucets all trace them. This is order hiding inside chaos, drawn as fine as spider silk. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/77d7a7c2-369d-492e-a58e-c55d3cb992e2/251204_22.00_00_15_03.Still002.jpg visuals Golden Seed A sphere of golden points holds a five-petaled heart, while dotted spirals unwind outward in every direction. Five-fold symmetry is forbidden in ordinary crystals, yet it turns up everywhere in living things, from apple cores to sunflower heads. Quasicrystals cheat the rule by borrowing order from a higher dimension. 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Project Five https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847418431-66X2HHF6ZNWKOC9VHLZI/20140301_Trade-151_0124-copy.jpg Portfolio 2 - Project Five https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847417789-TGDSX064I3PZELTJI8Q1/20140301_Trade-151_0124-copy.jpg Portfolio 2 - Project Five https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847417384-6R65UAAJW6EW4KUCUT6U/20140301_Trade-151_0124-copy.jpg Portfolio 2 - Project Five https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847416982-R2067X6OPKY3MG304FSJ/20140301_Trade-151_0124-copy.jpg Portfolio 2 - Project Five https://www.billnoll.com/portfolio-2/project-four-yjynj-nsxbr monthly 0.5 2022-05-24 https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847905386-9EM9GLHE12726EXM68S4/image-asset.jpeg https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847906050-Q37PFW075QZ6YU9179YP/image-asset.jpeg https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847907929-YZFF0BD112Q9PF2G1BK6/image-asset.jpeg https://www.billnoll.com/portfolio-2/project-three-8zgh7-a73rj monthly 0.5 2022-05-24 https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847412609-54WSVSAFRMKZEQTXDKB9/20140301_Trade-151_0124-copy.jpg Portfolio 2 - 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