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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 Invariant Torus When a system moves with two frequencies whose ratio is irrational, its path never closes. It winds forever around the surface of a torus, eventually passing arbitrarily close to every point. The KAM theorem, developed by Kolmogorov, Arnold and Moser between 1954 and 1963, showed that many such tori survive small disturbances, explaining why planetary orbits remain stable. The same geometry now guides fusion research: stellarators such as Wendelstein 7-X confine plasma on nested magnetic tori, and their success may decide whether fusion power becomes practical. 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/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/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/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/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/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/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/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/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 The layers of a neural network between input and output are called hidden layers. A 1989 theorem showed that even a single hidden layer can, in principle, approximate any continuous function. What these layers actually compute in practice is largely unknown. The young field of mechanistic interpretability attempts to read their internal circuits directly. Success would make it possible to verify the reasoning of artificial systems rather than only observe their outputs. 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 In 1963 Edward Lorenz found that a simple model of convection never settles into a repeating cycle, yet never leaves a bounded region. Its trajectories trace a strange attractor, a set with fractional dimension. The consequence is sensitive dependence on initial conditions: weather becomes unpredictable beyond about two weeks regardless of computing power. Whether machine-learned forecasting can approach that theoretical limit, or reveal structure hidden within it, is now being actively tested. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/69e5663a-ce4c-4856-a89f-c7d1ee4eb02c/Screenshot+2026-10-08+at+13.03.23.png Visuals Phase-Space Flow A mechanical system can be described as a single point moving through phase space, whose coordinates are every position and momentum at once. Liouville showed in 1838 that as a region of such points evolves, it may stretch and fold into long filaments like these, but its volume never changes. That principle underlies statistical mechanics. Modern simulations built to respect it suggest a small chance that Mercury's orbit becomes unstable within the Sun's lifetime, and long-term solar system stability remains an open question. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/8ad0b488-5499-4654-b163-e6ad6d51dd31/Screenshot+2026-10-08+at+13.04.12.png Visuals Spherical Harmonics Any pattern on a sphere can be decomposed into spherical harmonics, the natural vibration modes of a spherical surface. The Sun rings in millions of such modes, allowing helioseismologists to map its interior from surface oscillations alone. The same decomposition is applied to the cosmic microwave background. Next-generation observatories are now searching those maps for a faint twisting pattern that would be direct evidence of gravitational waves from the first instant after the Big Bang. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/0f2becb8-f70d-4280-a878-7d2d892086e8/Screenshot+2026-10-08+at+13.10.21.png Visuals Moiré When two families of fine lines overlap at a slight angle, a larger pattern emerges that neither family contains: the moiré, visible here as concentric rings at the center. In 2018 physicists stacked two sheets of graphene with a twist of about 1.1 degrees and found that the resulting moiré lattice turned the material superconducting. The discovery opened the field of twistronics, in which the angle between atomic layers becomes a design parameter for entirely new materials. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/6c942593-c827-4667-a0b7-40b5ba20e99b/Screenshot+2026-10-08+at+13.12.01.png Visuals Flow Past an Obstacle Streamlines divide and rejoin around a central obstruction, the classical problem of flow past a cylinder. Depending on speed and viscosity, the wake may remain smooth or shed an alternating trail of vortices, analyzed by Theodore von Kármán in 1911. All such behavior follows from the Navier-Stokes equations. Whether those equations always have smooth solutions in three dimensions is unproven, and it is one of the seven Millennium Prize Problems, carrying a one-million-dollar award. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/40388998-1311-4767-a8a0-7da7172fb2f3/Screenshot+2026-10-08+at+13.13.19.png Visuals Diffraction When waves pass through a regular structure, they interfere to produce a pattern of bright rings and radiating streaks that encodes the structure itself. X-ray diffraction revealed the arrangement of atoms in crystals and, through Rosalind Franklin's 1952 photograph, the double helix of DNA. Modern X-ray free-electron lasers produce pulses short enough to capture diffraction from single molecules before they are destroyed, moving toward filming chemical reactions as they happen. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/a85713d8-1168-45eb-946c-e8674859b115/Screenshot+2026-10-08+at+13.13.51.png Visuals Pursuit Curves Place several points at the corners of a polygon and let each move steadily toward its neighbor. The polygon rotates and contracts, and each point traces a spiral inward to the center. Pursuit problems of this kind were studied by Pierre Bouguer in 1732 and later became central to navigation and guidance. Today the same mathematics informs the decentralized control of robot swarms, where simple local rules must produce reliable collective behavior. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/a08c1f19-b6f1-4428-a617-6c31418c3a7b/Screenshot+2026-10-08+at+13.17.22.png Visuals Chaotic Mixing The marbled bands show a fluid that has been repeatedly stretched and folded. In 1984 Hassan Aref showed that even slow, orderly flows can mix chaotically, a process called chaotic advection, which now guides the design of microfluidic devices. Fully developed turbulence is harder still. Richard Feynman called it the most important unsolved problem of classical physics, and a complete predictive theory of turbulence remains one of the outstanding goals of mathematical physics. https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/b737983e-f0b2-4540-9e18-2432ba4e9aae/Screenshot+2026-10-08+at+13.18.01.png Visuals Point at Infinity Every line in this figure passes through a single point, a configuration projective geometry calls a pencil of lines. Renaissance painters discovered that parallel lines in a scene must converge to such a point on the canvas, and in 1639 Girard Desargues formalized the idea: parallel lines meet at a point at infinity. Projective geometry now underlies computer vision. Roger Penrose's twistor theory goes further, proposing to rebuild spacetime itself from light rays, a program still in development. https://www.billnoll.com/trumpet daily 0.75 2024-11-04 https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755080740-XJT30SQMSM7A5ZWE5OIT/IMG_2361.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755078111-900KF91XMBKNG2YF423V/IMG_2346.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755080739-LJ1Q6BDGWI0RX0IGK0CX/IMG_2338.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730754832806-UVBNJL6BOJF6CKG2ZWVN/IMG_2310.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755087283-ZNXK8K001I7QYBMN8W7R/IMG_2320.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730754833290-3RV2OW1695N0OIKUZWLC/IMG_2308.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755084962-NEG8QUCSW95CAHTK7VJY/IMG_2314.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755083176-OLLG3U67AMXTXTBL1QIN/IMG_2313.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755078194-Q6ACIWA8AI0WZQKR4L61/IMG_2348.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730754834666-VNFGY0WI0M6WYSUZBHXT/IMG_2309.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730754830736-RSJ7ZPUW95SE6PHQT8G6/IMG_2311.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730754830738-2RQZHGTQTW7J3UET96EQ/IMG_2312.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755085036-YQCHY3NW51X127EPPE6T/IMG_2316.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755087200-C747245CVIMSL5ZE879J/IMG_2317.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755089222-FXIUD3F2IGXHZDBT4ULM/IMG_2325.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755089424-B2X79NLY4C5JL83WGZDI/IMG_2326.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755090901-BEZGT716NKGHDOU40YQW/IMG_2328.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755091721-HSYU96H48W7Y8FQBLHDY/IMG_2335.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755093872-N0ZZ75VZNXKZCL208DUG/IMG_2350.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755094303-7GMX488Y1L2WH5T2WLBJ/IMG_2311.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755096041-V8PJO61NK41GWEASL5XC/trumpet-vintage-silver.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755082763-5Q1AART27084MZDKJ46W/IMG_2368.jpg trumpet https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/1730755092551-J1ITAR4SW3FJP4HYDAKC/IMG_2365.jpg trumpet https://www.billnoll.com/car daily 0.75 2026-01-05 https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/7c65dbfa-5a27-41fd-a95d-490b057fe9f2/01.jpg https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/a753f122-d1fb-444f-a5be-6a6ca457cfe0/02.jpg https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/3b9cf597-f936-4c57-b3b2-add91a6654f6/3.jpg https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/82b61acc-813c-4e2e-b32b-8ff78d5b30c7/4.jpg https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/e6c048d3-2278-4e9d-9eef-c768a685c0d2/gary_view.jpg https://images.squarespace-cdn.com/content/v1/60975a9e2896e65339ef566b/d563a6c7-c5ae-4197-b183-869d0e325ea0/noll_view.jpg https://www.billnoll.com/wallpaper-soon daily 0.75 2026-08-25 https://www.billnoll.com/textiles-soon daily 0.75 2026-08-25 https://www.billnoll.com/portfolio-2 daily 0.75 2024-06-09 https://www.billnoll.com/portfolio-2/project-six-sz8wl-wd5dw monthly 0.5 2022-05-24 https://images.squarespace-cdn.com/content/v1/5ec321c2af33de48734cc929/1589847421455-8JW2CPL3ZB56F2Z8ACLN/20140301_Trade-151_0124-copy.jpg Portfolio 2 - 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