{ "$schema": "./document.schema.json", "schemaVersion": "2.0-draft", "status": "authored-draft", "id": "strange-shapes", "title": "Strange Shapes", "language": "en-GB", "description": "Twenty independent mathematical encounters with circles, triangles, spheres, tori and the Möbius strip.", "playbackPolicy": { "layoutOwner": "player", "axes": ["topic", "passage-within-topic"], "rangeDegrees": [0, 360], "circular": true, "allocation": "random-with-spacing-on-edition-load", "stableFor": "loaded-edition-session", "reallocateOnPowerToggle": false, "receptionGaps": "required-between-stations-on-both-axes", "maxAudiblePassages": 1 }, "topics": [ { "id": "circle", "title": "A Curve Around a Centre", "sourceIds": ["wikidata-circle"], "passages": [ { "id": "circle-tether", "title": "A pencil on a tether", "text": "Imagine a pencil attached to a fixed point by a taut thread. Moving the pencil around that point draws a circle. Every pencil position stays the same distance from the centre. A simple restriction on distance becomes a complete, closed curve." }, { "id": "circle-across", "title": "Two radii across", "text": "Draw a straight line across a circle through its centre. The centre divides that crossing into two equal lengths, each a radius. This whole crossing is the diameter. A circle with a radius of three centimetres therefore measures six centimetres straight across its centre." }, { "id": "circle-reflection", "title": "A fold through the middle", "text": "Imagine folding a drawing of a circle along a line through its centre. Points on the curve meet matching points on the opposite side of the fold. Both remain equally far from the centre, so neither half has a bulge without a partner." }, { "id": "circle-turn", "title": "Turning without changing", "text": "Place an unmarked circle on an imaginary turntable with its centre exactly over the pivot. Any turn leaves the curve in the same position. Rotation changes where each point has travelled, but preserves its distance from the centre, leaving the outline unchanged." } ] }, { "id": "triangle", "title": "Three Sides Meet", "sourceIds": ["wikidata-triangle"], "passages": [ { "id": "triangle-three-points", "title": "Closing the third side", "text": "Place three points on a sheet of paper, keeping them off a single straight line. Join each pair with a straight segment. The result is a triangle: three sides enclosing a region. The third segment closes the opening left by the first two." }, { "id": "triangle-impossible-sides", "title": "A gap that cannot close", "text": "Try making a triangle with rods measuring two, three and six centimetres. Even stretched straight, the two shorter rods reach only five centimetres. They cannot bridge the longer rod's endpoints. Three proposed side lengths do not always describe a triangle that can actually close." }, { "id": "triangle-midpoint", "title": "One corner, two triangles", "text": "Mark the midpoint of one side of a triangle, then connect that mark to the opposite corner. Two smaller triangles appear. Their bases have equal length, and their shared corner gives them equal height above those bases, so their areas are equal." }, { "id": "triangle-scale", "title": "A larger copy", "text": "Imagine enlarging a triangle until every side is twice its original length. Its height doubles too. Since its area depends on both base and height, the new triangle covers four times the area. Doubling a shape's lengths does more than double the space it encloses." } ] }, { "id": "sphere", "title": "The Surface of a Ball", "sourceIds": ["wikidata-sphere"], "passages": [ { "id": "sphere-surface", "title": "An outside with no thickness", "text": "In geometry, a sphere is the surface of a ball. Imagine an ideal shell with no thickness: there is room inside it, but that interior is not part of the sphere itself. The distinction separates a curved surface from the solid region it surrounds." }, { "id": "sphere-radius", "title": "One distance in every direction", "text": "Imagine a point held exactly five centimetres from a fixed centre, free to move in every direction through space. Its possible positions form a sphere. Moving closer to the centre puts it inside that surface; moving farther away places it outside." }, { "id": "sphere-slice", "title": "Circles through a shell", "text": "An imaginary flat sheet passing through the centre of a sphere meets its surface in a circle. Move the sheet away from the centre and that circle shrinks. At the sphere's farthest extent, the meeting becomes a single point before disappearing altogether." }, { "id": "sphere-coordinates", "title": "A point found by numbers", "text": "A sphere's equation can test whether a point belongs on its surface. From the centre, move three units east and four units north, with no upward movement. Three squared plus four squared equals twenty-five, placing that point on a sphere of radius five." } ] }, { "id": "torus", "title": "A Circle Sweeps a Ring", "sourceIds": ["wikidata-torus"], "passages": [ { "id": "torus-sweep", "title": "A circle travelling around", "text": "Imagine a vertical circle beside a vertical pole, with a clear gap between them. Carry the circle all the way around the pole while keeping its distance unchanged. The surface it sweeps resembles a doughnut. This ring-shaped surface is a torus." }, { "id": "torus-radii", "title": "Two distances shape a ring", "text": "Build an imaginary torus by revolving a circle of radius one around an axis three units from its centre. The inner rim lies two units from the axis; the outer rim lies four units away. One radius controls the tube, while another controls its circuit." }, { "id": "torus-routes", "title": "Two ways around", "text": "Imagine a tiny walker on a doughnut-shaped torus. One closed route circles the central opening. Another passes around the tube itself, through the opening and back over the outside. Both return the walker to the starting point, but wrap around different parts of the shape." }, { "id": "torus-middle-slice", "title": "Two circles in one plane", "text": "Picture a doughnut-shaped torus resting level on a table. An imaginary horizontal plane through the middle of its tube meets the surface along two circles. One follows the inner rim and the other the outer rim, leaving the central opening clear." } ] }, { "id": "mobius-strip", "title": "A Surface with One Side", "sourceIds": ["wikidata-mobius-strip"], "passages": [ { "id": "mobius-half-turn", "title": "Joining after a twist", "text": "Imagine joining the short ends of a paper strip after giving one end a half turn. The ideal surface represented by this model is a Möbius strip. That small twist produces a surface with one continuous side and a single boundary edge." }, { "id": "mobius-tracing", "title": "A line stays on the surface", "text": "Imagine drawing along the middle of a Möbius strip without lifting the pencil. The route reaches what initially looked like the paper's opposite face before eventually returning to its starting point. No edge has been crossed: the apparently separate faces belong to one continuous side." }, { "id": "mobius-boundary", "title": "Following the edge", "text": "A Möbius strip has a single boundary edge. In a paper model, the half turn joins one long edge of the original rectangle to the other. Following the boundary therefore visits both original long edges as parts of one uninterrupted circuit." }, { "id": "mobius-local-view", "title": "A small patch hides the twist", "text": "A small patch of a Möbius strip can look like an ordinary piece of ribbon. The surprising feature emerges when the whole surface is followed around its loop. A local glimpse does not reveal how distant parts connect to make one continuous side." } ] } ], "provenance": { "narrationOrigin": "Newly written AI-assisted narration for REVOUAR. The mathematical constructions, numerical examples and deductions are our own explanatory work, using the identified Wikidata structured facts as starting points.", "reuseScope": "CC0 structured entity facts only; no linked assets or source passages imported. The explanatory constructions are newly written, and no audio has been prepared.", "rightsLimitations": "CC0 applies to the cited structured data, not to images, recordings, linked websites or discussion pages. None of those assets are included. The source designation does not relicense REVOUAR software or branding.", "endorsement": "Source names identify factual references and imply no endorsement of this narration.", "sources": [ { "id": "wikidata-circle", "title": "Circle (Q17278)", "publisher": "Wikidata", "url": "https://www.wikidata.org/w/index.php?title=Q17278&oldid=2534027081", "checkedOn": "2026-09-16", "reuseBasis": "CC0 entity identity and description of a Euclidean curve; distance, reflection and rotation passages are our own mathematical constructions and deductions.", "policyUrls": ["https://www.wikidata.org/wiki/Wikidata:Copyright", "https://creativecommons.org/publicdomain/zero/1.0/"], "license": "CC0-1.0", "licenseUrl": "https://creativecommons.org/publicdomain/zero/1.0/", "scope": "Wikidata structured entity data only", "revision": 2534027081, "dataFields": ["labels.en", "descriptions.en"] }, { "id": "wikidata-triangle", "title": "Triangle (Q19821)", "publisher": "Wikidata", "url": "https://www.wikidata.org/w/index.php?title=Q19821&oldid=2544436770", "checkedOn": "2026-09-16", "reuseBasis": "CC0 identification of the three-sided shape; point, rod, midpoint and enlargement examples are our own mathematical constructions and deductions.", "policyUrls": ["https://www.wikidata.org/wiki/Wikidata:Copyright", "https://creativecommons.org/publicdomain/zero/1.0/"], "license": "CC0-1.0", "licenseUrl": "https://creativecommons.org/publicdomain/zero/1.0/", "scope": "Wikidata structured entity data only", "revision": 2544436770, "dataFields": ["labels.en", "descriptions.en"] }, { "id": "wikidata-sphere", "title": "Sphere (Q12507)", "publisher": "Wikidata", "url": "https://www.wikidata.org/w/index.php?title=Q12507&oldid=2532463522", "checkedOn": "2026-09-16", "reuseBasis": "CC0 surface description and defining equation; shell, moving-point, slice and coordinate examples are our own explanatory constructions and deductions.", "policyUrls": ["https://www.wikidata.org/wiki/Wikidata:Copyright", "https://creativecommons.org/publicdomain/zero/1.0/"], "license": "CC0-1.0", "licenseUrl": "https://creativecommons.org/publicdomain/zero/1.0/", "scope": "Wikidata structured entity data only", "revision": 2532463522, "dataFields": ["labels.en", "descriptions.en", "P2534"] }, { "id": "wikidata-torus", "title": "Torus (Q12510)", "publisher": "Wikidata", "url": "https://www.wikidata.org/w/index.php?title=Q12510&oldid=2517788891", "checkedOn": "2026-09-16", "reuseBasis": "CC0 description of a circle-generated surface of revolution and defining equation; pole, radius, walking and plane examples are our own explanatory constructions and deductions.", "policyUrls": ["https://www.wikidata.org/wiki/Wikidata:Copyright", "https://creativecommons.org/publicdomain/zero/1.0/"], "license": "CC0-1.0", "licenseUrl": "https://creativecommons.org/publicdomain/zero/1.0/", "scope": "Wikidata structured entity data only", "revision": 2517788891, "dataFields": ["labels.en", "descriptions.en", "P2534"] }, { "id": "wikidata-mobius-strip", "title": "Möbius strip (Q226843)", "publisher": "Wikidata", "url": "https://www.wikidata.org/w/index.php?title=Q226843&oldid=2541315422", "checkedOn": "2026-09-16", "reuseBasis": "CC0 identification of a surface with one side and one edge; paper, pencil, boundary and local-view passages are our own explanatory constructions and deductions.", "policyUrls": ["https://www.wikidata.org/wiki/Wikidata:Copyright", "https://creativecommons.org/publicdomain/zero/1.0/"], "license": "CC0-1.0", "licenseUrl": "https://creativecommons.org/publicdomain/zero/1.0/", "scope": "Wikidata structured entity data only", "revision": 2541315422, "dataFields": ["labels.en", "descriptions.en"] } ] } }