{"title":"Signature","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eThe Signature series forms the core of Axisophy - a growing body of mathematical visualisations and scientific infographics that reveal the structures underlying our world.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eThe range spans number theory, geometry, evolutionary biology, astronomy, and earth science. You'll find prime number patterns made visible in the Ulam and Sacks spirals, recursive fractal constructions like the Apollonian gasket, radial phylogenetic trees mapping the evolutionary relationships of hundreds of species, and data-rich infographics charting geological time and planetary systems.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eEach design begins with research - published datasets, peer-reviewed science, established mathematical theory - and translates it into clean, considered visual form. The emphasis is on clarity and proportion: designs that work as information and as objects on the wall.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003ePrinted on 250gsm archival matte paper through our global print network, the Signature series offers museum-quality printing with worldwide delivery. These are the prints that started Axisophy - and the foundation everything else builds from.\u003c\/p\u003e","products":[{"product_id":"ulam-spiral-minimal","title":"Ulam Spiral","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThis Ulam spiral art print maps every integer from 1 to 251,001 across a 501 × 501 grid, spiralling outward from the centre, with the 22,115 primes among them highlighted - roughly 8.8% of the grid. Twin-prime pairs, prime gaps and the famous diagonal alignments all surface in the pattern: structure emerging from a sequence usually treated as random.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThose diagonals are not an accident of drawing. Every line in the spiral corresponds to a quadratic polynomial, and some quadratics - Euler's n² + n + 41 is the best-known case - produce primes at far above the average rate. The spiral makes that fact visible: order and irregularity held in the same image.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eStanisław Ulam found the pattern in 1963, doodling through a tedious conference talk; Martin Gardner put it on the cover of Scientific American the following March, and it has remained the standard way of seeing the primes ever since.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092532897,"sku":"APW-USM-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092565665,"sku":"APW-USM-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ulam-spiral-white-print.jpg?v=1762261536"},{"product_id":"sacks-spiral-minimal","title":"Sacks Spiral","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThis Sacks spiral art print places every integer from 1 to 250,000 on a single spiral, each number set at a distance proportional to its square root, with one full turn per perfect square. The 22,044 primes among them - 8.82% of the total - are marked in black.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe square-root spacing is what makes the image work. Because every perfect square lands on the same ray, the primes fall into sweeping curves rather than the diagonals of a square spiral, and quadratics that are unusually rich in primes show up as long unbroken arcs. The print holds 2,588 twin-prime pairs, and a largest gap of 86 consecutive composite numbers.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eRobert Sacks devised the layout in 1994, working outward from the older Ulam spiral of 1963. Where Ulam used a square grid, Sacks used polar coordinates - the same primes, a different geometry, and a noticeably different picture of how they distribute.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092467361,"sku":"APW-SSM-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092500129,"sku":"APW-SSM-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-sacks-spiral-white-print.jpg?v=1762261420"},{"product_id":"deepest-trenches","title":"Deepest Trenches","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe ocean's ten deepest points, scaled by depth. The Challenger Deep in the Mariana Trench reaches about 10,935 metres below the surface - drop Everest into it and two kilometres of water would still close over the summit.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eEvery trench here marks a subduction zone, where one tectonic plate bends beneath another and the seafloor is dragged down with it. Most trace the rim of the Pacific, which is why the same boundaries produce the world's deepest water and its most active volcanoes and earthquakes.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eConditions at the bottom are extreme in a way that resists intuition: pressure exceeding a thousand atmospheres, no light at all, and temperatures a degree or two above freezing. Four people have visited the Challenger Deep floor; twelve have walked on the Moon.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe composition mirrors its companion print, The Highest Mountains, inverted - measurement turned into form, descending instead of rising. Part of the Signature series.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092401825,"sku":"APW-DTR-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092434593,"sku":"APW-DTR-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-deepest-trenches-white-print.jpg?v=1762261317"},{"product_id":"highest-mountains","title":"Highest Mountains","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eFourteen mountains on Earth rise above 8,000 metres. This print ranks all of them by elevation, from Everest at 8,849 metres down to Shishapangma at 8,027 - a spread of just 822 metres across the entire set.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThat narrowness is the point. The fourteen are not scattered across the planet but packed into a single arc of the Himalaya and Karakoram, spanning Nepal, China, Pakistan and India. Everything above 8,000 metres on Earth sits within about 2,000 kilometres of everything else above 8,000 metres, the product of one collision: the Indian plate driving into Eurasia some 50 million years ago, and still lifting the range by a few millimetres each year.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe threshold matters to climbers too. Above 8,000 metres is the death zone, where the air holds roughly a third of the oxygen available at sea level and the body deteriorates faster than it can recover, no matter how long it rests.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eDrawn as a continuous geometric range rather than fourteen separate profiles, the print reads as both an infographic and a landscape. Part of the Signature series.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092336289,"sku":"APW-GEO-HMT-P-500x700-AM-COL","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092369057,"sku":"APW-GEO-HMT-P-700x1000-AM-COL","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-highest-mountains-white-print.jpg?v=1762261247"},{"product_id":"apollonian-gasket","title":"Apollonian Gasket","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eStart with four circles, each touching the other three. In every curved gap between them, draw the largest circle that fits. Now there are more gaps, and each of those takes another circle. Repeat forever and the result is an Apollonian gasket: a surface of circles that never overlap, never quite fill the plane, and shrink towards infinity.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe curvatures behave remarkably. Descartes showed in 1643 that any four mutually tangent circles satisfy a fixed relationship between their curvatures - and a consequence is that if the first four are whole numbers, every circle in the gasket that follows is too, all the way down. The picture is geometry; the numbers underneath it are pure arithmetic.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eApollonius of Perga was working on tangent circles around 200 BCE, some two thousand years before the word fractal existed. The gasket has a Hausdorff dimension of about 1.3057: more than a line, less than a surface, which is a precise way of saying it sits between dimensions.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092270753,"sku":"APW-AG1-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092303521,"sku":"APW-AG1-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-apollonian-gasket-white-print.jpg?v=1762261137"},{"product_id":"fermats-spiral","title":"Fermat’s Spiral","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eFermat's spiral grows outward as the square root of the angle: each successive turn sits a little closer to the last than the one before it. Two arms wind from the centre in exact opposition, each completing twenty turns.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThat square-root growth has a consequence worth knowing. Every full turn encloses the same additional area as the one before, so points spaced evenly along the curve are also spaced evenly across the surface. Nothing crowds at the centre; nothing thins at the rim.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eIt is why the shape turns up wherever things must pack without gaps or overlaps. Sunflower florets, pine cone scales and daisy heads all approximate it - a fact formalised in 1979, when Helmut Vogel showed that a Fermat spiral with a fixed angular step reproduces the sunflower head almost exactly. Pierre de Fermat had described the curve in 1636, three centuries before anyone noticed the plants were using it.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092205217,"sku":"APW-FS1-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45864092237985,"sku":"APW-FS1-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-fermats-spiral-white-print.jpg?v=1762261063"},{"product_id":"morse-code","title":"Morse Code","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe International Morse Code alphabet: the twenty-six letters and ten digits, set as dots and dashes. No punctuation, no accented characters - the core operational set as standardised by the ITU.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe system runs entirely on proportion. A dot is one unit; a dash is three. The gap between elements within a letter is one unit, between letters three, between words seven. Nothing else is specified - not pitch, not medium, not speed - which is why the same code works as a tone, a flash of light, or a tap on a hand.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eSamuel Morse and Alfred Vail built it in 1836 around a simple economy: the commonest letters get the shortest codes. E is a single dot, T a single dash, while rarer letters like Q and Y run to four elements each. Vail is said to have counted the type in a printer's case to work out which letters English used most. It was variable-length encoding a century before information theory gave it a name.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864092172449,"sku":"APW-MSC-P-500x700-AM-COL","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-morse-code-print.jpg?v=1762260952"},{"product_id":"insecta-phylogenetic-tree-500-mono","title":"Insecta Phylogenetic Tree - 500 - Mono","description":"\u003cp\u003eInsects account for more than half of all described living species, which makes any diagram of them a sample rather than a survey. This one maps 500, arranged by descent across roughly 400 million years - since before there were trees to climb or flowers to visit.\u003c\/p\u003e\n\u003cp\u003eThe beetles are the problem the diagram makes visible. There are around 400,000 described species of them, more than in any other order of any kind of animal, and they crowd one section of the tree in a way that no amount of drawing can make proportionate. J.B.S. Haldane's remark about the Creator having an inordinate fondness for beetles was a comment on exactly this shape.\u003c\/p\u003e\n\u003cp\u003eFlight appears early and only once. Every winged insect on the diagram descends from a single ancestor that solved the problem well before vertebrates managed it, and the groups that lack wings today - fleas, lice, worker ants - mostly lost them again rather than never having had them.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of scientific illustration prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864090566817,"sku":"AXS-IPT-MONO-500-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Insecta-Mono-PhylogeneticTree-print.jpg?v=1762262848"},{"product_id":"dinosauria-phylogenetic-tree-500-mono","title":"Dinosauria Phylogenetic Tree - 500 - Mono","description":"\u003cp\u003eFive hundred dinosaur species across 165 million years, arranged by descent rather than by date. The diagram makes one point immediately, and it is not the one most people expect: birds are not descended from dinosaurs in the way that mammals are descended from reptiles. Birds are dinosaurs. The lineage never ended; it just took to the air.\u003c\/p\u003e\n\u003cp\u003eThe split at the base is the deepest one in the group. Saurischians - the sauropods and theropods - and ornithischians diverged early and then developed almost every subsequent form independently of one another. Convergence is everywhere in the branches: armour, beaks, herbivory and bipedalism each appear more than once, on lineages that had already been separate for tens of millions of years.\u003c\/p\u003e\n\u003cp\u003eDinosaur phylogeny is also the least settled of these diagrams. The relationships here follow current palaeontological consensus, and parts of that consensus have been rearranged more than once in the last decade as new specimens have emerged. What the tree shows is the best available reading of a record that is still being excavated.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of scientific illustration prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864090534049,"sku":"AXS-DPT-MONO-500-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Dinosauria-Mono-PhylogeneticTree-mockup-1-flat-crop.jpg?v=1762263100"},{"product_id":"mammalia-phylogenetic-tree-500-mono","title":"Mammalia Phylogenetic Tree - 500 - Mono","description":"\u003cp\u003eFive hundred mammal species arranged by descent, from blue whales to shrews. Every one of them traces back to a small, warm-blooded animal that lived through the extinction event 66 million years ago - and the explosive fan of branches near the base of the diagram is the record of what happened once the dinosaurs were gone and the world was empty.\u003c\/p\u003e\n\u003cp\u003eSome of the closest relationships are the least intuitive. Whales sit inside the group that contains hippos, cattle and deer, because that is where the genetics and the fossil record both put them. Elephants are closer to manatees than to any large land animal. Bats are not close to birds, or to rodents, or to anything else that seems obvious.\u003c\/p\u003e\n\u003cp\u003eRodents and bats between them account for well over half of all mammal species, which the tree shows plainly as two dense thickets among otherwise sparse branches. The animals people picture when they hear the word mammal are, statistically, the exceptions.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of scientific illustration prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864090435745,"sku":"AXS-MPT-MONO-500-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Mammalia-Mono-PhylogeneticTree-print.jpg?v=1762263306"},{"product_id":"aves-phylogenetic-tree-500-mono","title":"Aves Phylogenetic Tree - 500 - Mono","description":"\u003cp\u003eFive hundred bird species arranged by descent, radiating outward from a single common ancestor. Distance from the centre is time; every fork is a moment when one lineage became two. Hummingbirds and ostriches sit on the same diagram, separated by roughly 100 million years of divergence.\u003c\/p\u003e\n\u003cp\u003eThe shape carries information the list of names cannot. Some branches fan wide and late - the passerines, which account for more than half of all living birds, erupt near the outer edge in a burst of recent speciation. Others run long and almost bare: the ostriches and their relatives split early and never diversified in the same way.\u003c\/p\u003e\n\u003cp\u003eThe arrangement follows current molecular phylogenetics rather than the older classifications built on anatomy, and the two disagree more often than is comfortable. Falcons turn out to be closer to parrots than to hawks. Flamingos are closest to grebes. The tree is drawn from what the genomes say, not from what the birds look like.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of scientific illustration prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864090304673,"sku":"AXS-APT-MONO-500-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Aves-Mono-PhylogeneticTree-print.jpg?v=1762263467"},{"product_id":"geological-timescale-recursive","title":"Geological Timescale - Recursive","description":"\u003cp\u003eEarth is about 4.5 billion years old, and almost nothing recognisable happens for the first four of them. That is the problem with drawing geological time to scale: the interesting part is a sliver at the end, and the sliver contains everything anyone has ever heard of.\u003c\/p\u003e\n\u003cp\u003eThis print solves it recursively. Each successive panel magnifies the tail of the one before, so the eye can move from the formation of the planet down to the Holocene without the recent losing all of its detail or the ancient losing all of its length. The Precambrian occupies almost 90 per cent of Earth's history and can still be shown alongside the 66 million years since the dinosaurs.\u003c\/p\u003e\n\u003cp\u003eWhat the arrangement exposes is how compressed the familiar is. Complex animals appear only in the last 600 million years. The whole age of the dinosaurs takes up less room than the emptiness preceding it. Written human history does not register at any scale on which the Cambrian is visible.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of scientific illustration prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45864090271905,"sku":"AXS-GTSR-COL-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Geological-Timescale-print.jpg?v=1762263652"},{"product_id":"e8-coxeter-projection","title":"E8: Coxeter Projection","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eE8 is the largest of the five exceptional root systems - a collection of 240 vectors in 8-dimensional space, arranged with a degree of symmetry that has no parallel in lower dimensions. It appears in string theory, in the classification of simple Lie algebras, and in the construction of the densest possible sphere packing in 8 dimensions. Mathematicians have been studying it for over a century and it continues to surprise them.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eThis print shows E8 projected onto its Coxeter plane - a canonical 2D view that preserves the full 30-fold rotational symmetry of the system. Each of the 240 roots falls into one of 8 concentric rings of 30 points. The 6,720 edges connect nearest-neighbour root pairs, each root touching exactly 56 others. Nothing is approximate or artistic - every position, every connection is mathematically exact.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eIn 2016 Maryna Viazovska proved that the E8 lattice gives the densest possible packing of spheres in eight dimensions - a problem open since Kepler posed its three-dimensional version in 1611. It won her the Fields Medal. The arrangement in this print is that packing, seen from the one angle that shows all of its symmetry at once.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45891882844321,"sku":"APW-E8C-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45891882877089,"sku":"APW-E8C-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-E8-Coxeter-mockup-1.jpg?v=1775494565"},{"product_id":"e7-coxeter-projection","title":"E7: Coxeter Projection","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eE7 is the second-largest of the five exceptional root systems, sitting between E8 and E6 in a family of objects with no straightforward lower-dimensional analogue. It has 126 roots in 7-dimensional space, each one connected to 32 others at the nearest-neighbour distance, giving 2,016 edges in total.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eThe Coxeter plane projection collapses all 7 dimensions into a single plane while preserving the system's 18-fold rotational symmetry. The 126 roots arrange themselves into 7 concentric rings of 18 - a structure that is genuinely visible in the print in a way that E8, with its greater density, makes harder to read. If E8 is the overwhelming one, E7 is the one you can actually study.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eE7 also carries a quiet piece of trivia: its 126 roots and 2,016 edges make it the exceptional system most often used to test conjectures, precisely because it is large enough to be interesting and small enough to compute. Where E8 is studied, E7 is worked with.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45891939729569,"sku":"APW-E7C-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45891939762337,"sku":"APW-E7C-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-E7-Coxeter-mockup-1.jpg?v=1775495758"},{"product_id":"f4-coxeter-projection","title":"F4: Coxeter Projection","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eF4 is the most unusual of the five exceptional root systems. Unlike E8, E7, and E6, it is non-simply-laced - meaning its 48 roots come in two distinct lengths, 24 long and 24 short, with a ratio of √2 between them. This asymmetry makes F4 structurally richer than its root count suggests, and gives it a character quite different from the other exceptional systems.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eIn the Coxeter plane projection, the 48 roots arrange into 4 concentric rings of 12, with 12-fold rotational symmetry throughout. All 1,128 possible connections between roots are shown - a choice that makes sense for F4, where the nearest-neighbour edge set alone gives an unusually sparse result owing to the mixed root lengths. The full connectivity reveals the geometry more completely.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003eThe two root lengths are not a defect but a signature. F4 is the symmetry group of the exceptional Jordan algebra, a 27-dimensional structure built from octonions that sits at the edge of what algebra permits - and the mismatch between long and short roots is what allows it to exist at all.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal\" dir=\"ltr\"\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (50 × 70 cm | 20 × 28 in)","offer_id":45891940188321,"sku":"APW-F4C-P-500x700-AM-WHT","price":50.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (70 × 100 cm | 28 × 40 in)","offer_id":45891940221089,"sku":"APW-F4C-P-700x1000-AM-WHT","price":80.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/AxisophyF4-Coxeter-mockup-1.jpg?v=1775495920"},{"product_id":"conformal-map-cm001","title":"Conformal Map CM001","description":"\u003cp\u003eCM001 spreads the grid into a six-petalled form with sharply pointed lobes alternating with curved scallops. Six lines of compression radiate from the centre, where dense bands mark the regions of steepest distortion. The petal tips show the function at its most expansive, the original grid stretched outward into the surrounding white space.\u003c\/p\u003e\n\u003cp\u003eA conformal map distorts the complex plane while preserving angles. Take a regular square grid, apply the function point by point, and the grid emerges curved, stretched and folded - yet every intersection still crosses at 90 degrees. Shape is abandoned; angle survives.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46163870023841,"sku":"AXS-CM001-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46163870056609,"sku":"AXS-CM001-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM001-crop-mockup.jpg?v=1777640195"},{"product_id":"conformal-map-cm002","title":"Conformal Map CM002","description":"\u003cp\u003eCM002 compresses the grid into a bulging cube-shaped envelope, its top and bottom near-flat, its sides curved outward, its corners softly squared. Eight focal points sit on the faces - four around the equator, four above and below - each surrounded by concentric ring structures where the function maps neighbourhoods of the plane down to a point. The interior is a dense, even weave; the visible structure lives at the faces and edges where compression peaks.\u003c\/p\u003e\n\u003cp\u003eConformal maps preserve angles everywhere except at their critical points, where the derivative vanishes. There the rule breaks: angles are multiplied rather than kept, and the grid wraps around itself. Those points are the ones the eye finds first, because they are where the geometry gives way.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46164133609633,"sku":"AXS-CM002-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46164133642401,"sku":"AXS-CM002-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM002-crop-mockup.jpg?v=1777640790"},{"product_id":"conformal-map-cm003","title":"Conformal Map CM003","description":"\u003cp\u003eCM003 takes the form of two pointed ellipses overlapping at right angles, creating a soft four-cornered envelope. The dense central disc is where both ellipses share coverage; the lighter outer regions show where only one ellipse extends. The aggregate texture is unusually flat - a near-uniform grey field - with structure visible mainly at the curved boundaries where the ellipses terminate.\u003c\/p\u003e\n\u003cp\u003eRiemann proved in 1851 that any simply connected region of the plane, however irregular, can be conformally mapped onto any other. A square becomes a disc, a disc becomes a crescent, and the angles survive the journey. The set of available transformations is, in effect, unlimited.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46164757610657,"sku":"AXS-CM003-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46164757643425,"sku":"AXS-CM003-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM003-crop-mockup.jpg?v=1777641375"},{"product_id":"conformal-map-cm004","title":"Conformal Map CM004","description":"\u003cp\u003eCM004 is the most structurally complex print in the series. The grid resolves into a layered system of concentric rings, with additional features extending outward from the main circular form into pointed and curved protrusions. Different regions carry markedly different densities, producing alternating bands of light and dark. The overall composition is fourfold symmetric, but the internal organisation rewards close inspection rather than reducing to a single readable form.\u003c\/p\u003e\n\u003cp\u003eEvery conformal map is a holomorphic function with a non-vanishing derivative, which means that under sufficient magnification it does only two things: rotate and scale. All the apparent complication of a print like this one is that simple local rule, applied consistently across the plane and compounded.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46164813512865,"sku":"AXS-CM004-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46164813545633,"sku":"AXS-CM004-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM004-crop-mockup.jpg?v=1777641762"},{"product_id":"conformal-map-cm006","title":"Conformal Map CM006","description":"\u003cp\u003eCM006 takes the form of two intersecting lenses, oriented horizontally and vertically, producing a four-cornered envelope with a dense black core at the centre. The compression toward this central singularity is smooth and continuous - the grid darkens from the periphery inward in a tight gradient. At the four corners, the grid thins to its sparsest, lightest weave.\u003c\/p\u003e\n\u003cp\u003eWhere a conformal map sends a whole neighbourhood of the plane towards a single value, the grid collapses inward and the transformation is no longer invertible. These singularities are the structural events in an otherwise smooth surface - the points around which everything else arranges itself.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46164815478945,"sku":"AXS-CM006-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46164815511713,"sku":"AXS-CM006-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM006-crop-mockup.jpg?v=1777641957"},{"product_id":"conformal-map-cm007","title":"Conformal Map CM007","description":"\u003cp\u003eCM007 sits inside a soft square envelope with four small protrusions at the diagonals. The interior carries a ripple pattern of concentric rings - more numerous and more closely spaced than the related CM008 - radiating from a central node. Each ring marks a circular zone where the function changes its rate of compression. The overall texture is smooth, with the rings themselves providing the only sharp edges in the composition.\u003c\/p\u003e\n\u003cp\u003eThe rings in a print like this are level curves: paths along which the transformation compresses the plane by a constant factor. They are not drawn. They emerge from the spacing of the grid alone, which is why they read as ripples rather than as lines.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46164901298337,"sku":"AXS-CM007-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46164901331105,"sku":"AXS-CM007-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM007-crop-mockup.jpg?v=1777642148"},{"product_id":"conformal-map-cm008","title":"Conformal Map CM008","description":"\u003cp\u003eCM008 applies a single transformation to a grid of 500 lines in each direction. The result has a nested circular structure with pinched, concave outer edges - regions where the function compresses the plane heavily, regions where it stretches it thin. The right-angle property holds throughout, even where the grid becomes too dense to resolve individual lines.\u003c\/p\u003e\n\u003cp\u003eThe idea was formalised by Riemann in 1851, and it turned out to matter well beyond pure mathematics. The same transformations describe fluid flow around an obstacle, the field around a charged conductor, and the stresses in a loaded plate - problems made tractable by bending the plane into a shape where the answer is obvious.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46165062353057,"sku":"AXS-CM008-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46165062385825,"sku":"AXS-CM008-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM008-crop-mockup.jpg?v=1777642323"},{"product_id":"conformal-map-cm009","title":"Conformal Map CM009","description":"\u003cp\u003eCM009 takes the form of a twisted, warped square - its edges curved into cusps, its corners softly rotated out of alignment. Several focal points sit across the interior, each surrounded by tight concentric rings where the function maps neighbourhoods of the plane to single values. The overall surface is a soft, even grey, with the focal points providing the only strong features. The asymmetric placement gives the print a sense of slow rotational drift.\u003c\/p\u003e\n\u003cp\u003eMultiplying by a complex number rotates and scales at once, and a conformal map applies that operation continuously across the plane, with the amount of rotation varying from point to point. The drift visible in this print is that varying argument made legible - the plane turning by different amounts in different places.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46165198995617,"sku":"AXS-CM009-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46165199028385,"sku":"AXS-CM009-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-ConformalMap-CM009-crop-mockup.jpg?v=1777642460"},{"product_id":"ulam-spiral-651","title":"Ulam Spiral 651","description":"\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eIn 1963, the Polish mathematician Stanisław Ulam sketched numbers in a square spiral during a tedious lecture and noticed something unexpected: the prime numbers tended to fall along diagonal lines. The pattern wasn't a fluke. The Ulam spiral has been studied ever since as one of the strangest visual phenomena in number theory - a hint of structure inside a sequence usually treated as essentially random.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eUlam Spiral 651 maps every integer from 1 to 423,801 across a square grid, with the 35,684 primes marked as black dots. The spiral begins at the centre with 1 and expands outward in the classic right-up-left-down rotation. At this density - roughly 8.4% of the grid is prime - the famous diagonal alignments are less visible as discrete lines and more present as a kind of grain across the surface, with the distribution thinning gradually toward the corners as primes become rarer at higher numbers.\u003c\/p\u003e\n\u003cp class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"\u003eThe result is a print that reads at two distances. Close up, individual diagonals and clusters reveal themselves. From across the room, the whole image becomes a textured field - a portrait of the primes as a landscape rather than a list.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46165454454945,"sku":"AXS-UL651-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46165454487713,"sku":"AXS-UL651-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/Axisophy-Ulam-651-crop-mockup.jpg?v=1777642803"},{"product_id":"gaussian-primes-521","title":"Gaussian Primes 521","description":"\u003cp\u003eOrdinary primes live on a line. Gaussian primes live on a plane. Take the complex integers - every number of the form a + bi where a and b are whole - and the question of which are prime has a two-dimensional answer.\u003c\/p\u003e\n\u003cp\u003eThe rules are not what you would expect. Two is no longer prime here; nor is five, or thirteen, or any prime that is one more than a multiple of four. Each of those splits into a pair of Gaussian primes, because each can be written as the sum of two squares. What survives intact are the primes three more than a multiple of four - 3, 7, 11, 19 - which cannot be split and stay prime in the complex plane.\u003c\/p\u003e\n\u003cp\u003eThis print maps every complex integer from -260 to +260 on both axes, 271,441 of them, and marks the 33,648 that are prime. Only 116 sit on the axes themselves. Colour follows distance from the origin, cycling through the spectrum as the shells of primes move outward, so the eye reads the radial structure before it reads the individual points.\u003c\/p\u003e\n\u003cp\u003eThe symmetry is exact and unavoidable: multiplying a Gaussian prime by -1, i or -i gives another Gaussian prime, so the whole field repeats eightfold about the origin. Whatever appears in one octant appears in all of them.\u003c\/p\u003e\n\u003cp\u003eOne question about this picture remains open. If you could step only on Gaussian primes, taking strides of some fixed maximum length, could you walk from the origin out to infinity? Nobody knows. It is called the Gaussian moat problem, and computer searches have got some way out before running into water.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Spectrum \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875844182177,"sku":"AXS-GAU521-S-700x700-AM-WHT-SPECTRUM","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Spectrum \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875844214945,"sku":"AXS-GAU521-S-1000x1000-AM-WHT-SPECTRUM","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875845296289,"sku":"AXS-GAU521-S-700x700-AM-WHT-DUSK","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875845361825,"sku":"AXS-GAU521-S-1000x1000-AM-WHT-DUSK","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875845329057,"sku":"AXS-GAU521-S-700x700-AM-WHT-MONO","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875845394593,"sku":"AXS-GAU521-S-1000x1000-AM-WHT-MONO","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-gaussian-primes-521-art-print-spectrum-700mm-mockup.jpg?v=1786624008"},{"product_id":"hurwitz-primes-521","title":"Hurwitz Primes 521","description":"\u003cp\u003eQuaternions extend the complex numbers into four dimensions: a + bi + cj + dk, with three separate square roots of minus one. Hurwitz integers are the points of that space where all four coordinates are whole numbers, or all four are halves - a lattice denser than the obvious one, and the right setting for arithmetic.\u003c\/p\u003e\n\u003cp\u003eA Hurwitz integer is prime when its norm - the sum of the squares of its four coordinates - is an ordinary prime. That single rule produces a result the Gaussian case does not. Lagrange proved in 1770 that every whole number is the sum of four squares, so every ordinary prime is a norm, and every ordinary prime therefore splits here. Nothing stays intact. Three, seven and eleven survive the move into the complex plane; none of them survives the move into the quaternions.\u003c\/p\u003e\n\u003cp\u003eMultiplication is the price. Quaternions do not commute - ij and ji point in opposite directions - so factorisation into primes is unique only up to reordering and recombination. Arithmetic still works, but it stops behaving politely.\u003c\/p\u003e\n\u003cp\u003eThis print takes a two-dimensional slice through that four-dimensional lattice, holding the remaining coordinates at a fixed half-integer point, and marks the 44,489 primes in a 521 × 521 field. Colour follows distance from the origin, cycling through the spectrum as the shells move outward.\u003c\/p\u003e\n\u003cp\u003eThere are twenty-four units in this system - the Hurwitz integers of norm one - and they sit at the vertices of the 24-cell, one of the six regular polytopes in four dimensions. Those same twenty-four points generate the F4 root system, which is why this print and the F4 Coxeter Projection are two views of the same underlying object.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Spectrum \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920400545,"sku":"AXS-HUR521-S-700x700-AM-WHT-SPECTRUM","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Spectrum \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920433313,"sku":"AXS-HUR521-S-1000x1000-AM-WHT-SPECTRUM","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920466081,"sku":"AXS-HUR521-S-700x700-AM-WHT-DUSK","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920498849,"sku":"AXS-HUR521-S-1000x1000-AM-WHT-DUSK","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920531617,"sku":"AXS-HUR521-S-700x700-AM-WHT-MONO","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920564385,"sku":"AXS-HUR521-S-1000x1000-AM-WHT-MONO","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-hurwitz-primes-521-art-print-spectrum-700mm-mockup.jpg?v=1786630135"},{"product_id":"octavian-primes-521","title":"Octavian Primes 521","description":"\u003cp\u003eThe octonions are the last stop. Real numbers, then complex, then quaternions, then octonions - eight dimensions - and then nothing. Hurwitz proved in 1898 that these four are the only number systems in which division works at all, so the sequence does not continue.\u003c\/p\u003e\n\u003cp\u003eEach extension costs something. The complex numbers give up ordering; the quaternions give up commutativity; the octonions give up associativity, which means that for octonions a, b and c, the products (ab)c and a(bc) are generally different numbers. Arithmetic survives this, but only just.\u003c\/p\u003e\n\u003cp\u003eThe octavian integers - Coxeter described them in 1946, building on Dickson's earlier work - form a lattice within that eight-dimensional space, and one of their points is prime when the sum of the squares of its eight coordinates is an ordinary prime. This print takes a two-dimensional slice through the lattice, holding the remaining coordinates at a fixed point, and marks the 22,572 primes in a 521 × 521 field. Colour follows distance from the origin, cycling through the spectrum as the shells move outward.\u003c\/p\u003e\n\u003cp\u003eIt is the sparsest of the three prime lattices, at 8.3 per cent against the Gaussian 12.4 and the Hurwitz 16.4, and the difference is visible at a glance when the three are hung together.\u003c\/p\u003e\n\u003cp\u003eThere are 240 units in this system, and they are the 240 roots of E8 - the largest of the five exceptional root systems and the densest sphere packing possible in eight dimensions. This print and the E8 Coxeter Projection describe the same lattice from different directions: one shows its symmetry, the other its primes.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Spectrum \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920629921,"sku":"AXS-OCT521-S-700x700-AM-WHT-SPECTRUM","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Spectrum \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920662689,"sku":"AXS-OCT521-S-1000x1000-AM-WHT-SPECTRUM","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920695457,"sku":"AXS-OCT521-S-700x700-AM-WHT-DUSK","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Dusk \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920728225,"sku":"AXS-OCT521-S-1000x1000-AM-WHT-DUSK","price":120.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ Large (70 × 70 cm | 28 × 28 in)","offer_id":46875920760993,"sku":"AXS-OCT521-S-700x700-AM-WHT-MONO","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"Mono \/ XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46875920793761,"sku":"AXS-OCT521-S-1000x1000-AM-WHT-MONO","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-octavian-primes-521-art-print-spectrum-700mm-mockup.jpg?v=1786630231"},{"product_id":"modular-circles-mc0002","title":"Modular Circles MC0002","description":"\u003cp\u003eMark a number of points evenly around a circle and label them 0, 1, 2, and so on. From each point, draw a straight line to the point with twice its number, wrapping around when the count exceeds the total. That is the entire construction: no curves are drawn, only straight lines between marked points.\u003c\/p\u003e\n\u003cp\u003eA cardioid appears. Not an approximation of one - the envelope of those straight lines is exactly the heart-shaped curve that forms in the reflected light inside a coffee cup, and which bounds the main body of the Mandelbrot set. Nothing in the instructions suggests a curve should emerge at all.\u003c\/p\u003e\n\u003cp\u003eThe rule generalises. A multiplier of three produces a nephroid with two cusps, four produces three, and so on: the number of cusps is always one less than the multiplier. Each is an epicycloid, the curve traced by a point on a circle rolling around another circle, arrived at here by a route that involves no rolling and no circles beyond the first.\u003c\/p\u003e\n\u003cp\u003eThe construction is old enough to have been a schoolroom exercise with pins and thread, and simple enough to explain in a sentence. What it demonstrates is that curvature can be an emergent property of straight lines - that a shape can be present in a rule without being present in any of its parts.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883481649313,"sku":"AXS-MODC-0002-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883481682081,"sku":"AXS-MODC-0002-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0002-art-print-700mm-mockup.jpg?v=1786707341"},{"product_id":"modular-circles-mc0003","title":"Modular Circles MC0003","description":"\u003cp\u003eMark points evenly around a circle, number them, and join each one by a straight line to the point at three times its number, wrapping around when the count runs past the total. Every line is straight. No curve is drawn at any stage.\u003c\/p\u003e\n\u003cp\u003eA nephroid emerges: two cusps, kidney-shaped, the curve traced by a point on a circle rolling around another circle exactly twice its size. It is the shape of the bright caustic that forms when parallel light reflects off the inside of a cylinder, which is why it turns up in a coffee cup and in the reflections inside a wedding ring.\u003c\/p\u003e\n\u003cp\u003eThe relationship between multiplier and curve is exact and simple: the number of cusps is always one less than the multiplier. Two gives a cardioid with one cusp, three gives this, four gives three cusps, and the sequence continues indefinitely. Each is an epicycloid, arrived at without anything rolling.\u003c\/p\u003e\n\u003cp\u003eWhat the construction demonstrates is that curvature can be emergent. The cusps are not drawn, not calculated, and not present in any individual line. They are a property of the whole family of lines, and of nothing within it.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883866804385,"sku":"AXS-MODC-0003-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883866837153,"sku":"AXS-MODC-0003-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0003-art-print-700mm-mockup.jpg?v=1786708434"},{"product_id":"modular-circles-mc0004","title":"Modular Circles MC0004","description":"\u003cp\u003eNumber a set of points spaced evenly around a circle, then join each to the point at four times its number, counting round again whenever the total is exceeded. The construction uses straight lines exclusively.\u003c\/p\u003e\n\u003cp\u003eThree cusps appear. The curve is an epicycloid - the path traced by a point on the rim of a circle rolling around the outside of another - and this particular one, with three cusps, is close kin to the deltoid that Euler studied in 1745 while working on a problem in optics.\u003c\/p\u003e\n\u003cp\u003eThe rule holds throughout the series: cusps are always one fewer than the multiplier. What changes as the multiplier climbs is not the principle but the density. At four the structure is still legible as a single closed curve. Higher multipliers begin to fill the circle with overlapping envelopes until the individual shape gives way to texture.\u003c\/p\u003e\n\u003cp\u003eThe whole thing was once a schoolroom exercise done with pins and thread, which is worth remembering when looking at it: every intersection here could in principle have been placed by hand.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883867525281,"sku":"AXS-MODC-0004-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883867558049,"sku":"AXS-MODC-0004-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0004-art-print-700mm-mockup.jpg?v=1786708509"},{"product_id":"modular-circles-mc0005","title":"Modular Circles MC0005","description":"\u003cp\u003ePoints evenly spaced around a circle, each connected by a straight line to the point at five times its number. Nothing curved is drawn; the count simply wraps around whenever it passes the total.\u003c\/p\u003e\n\u003cp\u003eFour cusps form. By this point in the series the pattern governing the construction is unmistakable - the cusp count trails the multiplier by exactly one - but the reason is not obvious from the drawing, and the proof requires the calculus of envelopes rather than anything visible in the figure.\u003c\/p\u003e\n\u003cp\u003eThe secondary structure is what makes this one worth looking at closely. Beyond the main envelope, the lines cross in a fine lattice of their own, denser toward the centre and looser at the rim, produced by nothing more than the arithmetic of remainders.\u003c\/p\u003e\n\u003cp\u003eThe construction has no adjustable parameters beyond the multiplier and the number of points. Everything visible follows from those two numbers, which is an unusually small amount of input for the amount of structure that results.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883868704929,"sku":"AXS-MODC-0005-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883868737697,"sku":"AXS-MODC-0005-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0005-art-print-700mm-mockup.jpg?v=1786708585"},{"product_id":"modular-circles-mc0006","title":"Modular Circles MC0006","description":"\u003cp\u003eEach point on the circle is joined by a straight line to the point at six times its number, counting round whenever the total is passed. As with every print in this series, no curve is drawn at any stage.\u003c\/p\u003e\n\u003cp\u003eFive cusps appear in the envelope, following the rule that governs the whole family: one fewer than the multiplier. Six is where the character of the series begins to change. The main curve is still clear, but the interior is now busy with the secondary crossings that will come to dominate at higher multipliers.\u003c\/p\u003e\n\u003cp\u003eThose interior structures are not incidental. They are envelopes in their own right, formed by subsets of the lines that happen to share a relationship, and they carry information about the factors of the numbers involved. Multipliers that share divisors with the point count produce visibly different interiors from those that do not.\u003c\/p\u003e\n\u003cp\u003eEverything here follows from two numbers and a rule that can be stated in one sentence. The construction predates computers by a long way; only the precision is modern.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883874177185,"sku":"AXS-MODC-0006-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883874209953,"sku":"AXS-MODC-0006-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0006-art-print-700mm-mockup.jpg?v=1786708640"},{"product_id":"modular-circles-mc0017","title":"Modular Circles MC0017","description":"\u003cp\u003eThe construction is identical to the rest of the series: points spaced evenly around a circle, each joined by a straight line to the point at seventeen times its number, counting round whenever the total is exceeded.\u003c\/p\u003e\n\u003cp\u003eThe result is different in kind. Sixteen cusps still form, obeying the rule that holds throughout - one fewer than the multiplier - but at this density the envelope is no longer the first thing the eye finds. The interior crossings dominate, and the print reads as woven texture before it resolves into a curve.\u003c\/p\u003e\n\u003cp\u003eSeventeen being prime is part of why it looks like this. A multiplier sharing factors with the number of points produces interiors that repeat and simplify; a prime multiplier shares nothing, so the lines distribute without settling into smaller repeating groups. The texture is denser and less resolvable as a result.\u003c\/p\u003e\n\u003cp\u003eIt is the same rule as the cardioid at the start of the series, applied fifteen steps further along. Nothing has been added, and the difference between the two prints is entirely a matter of how far a single number has been pushed.\u003c\/p\u003e\n\u003cp\u003ePart of the Signature series of mathematical art prints.\u003c\/p\u003e","brand":"Axisophy","offers":[{"title":"Large (70 × 70 cm | 28 × 28 in)","offer_id":46883880632481,"sku":"AXS-MODC-0017-S-700x700-AM-WHT","price":70.0,"currency_code":"GBP","in_stock":true},{"title":"XLarge (100 × 100 cm | 40 × 40 in)","offer_id":46883880665249,"sku":"AXS-MODC-0017-S-1000x1000-AM-WHT","price":120.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/files\/axisophy-modular-circles-mc0017-art-print-700mm-mockup.jpg?v=1786708696"}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0483\/1546\/5889\/collections\/apw-signature-collection-1.jpg?v=1762433340","url":"https:\/\/axisophy.com\/collections\/signature.oembed?page=2","provider":"Axisophy","version":"1.0","type":"link"}