[{"data":1,"prerenderedAt":1051},["ShallowReactive",2],{"layer:shape-and-space:extend":3},{"layer":4,"contentHash":1027,"dependencyHashes":1028,"approval":1044,"releaseId":1050},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1022,"reviewStatus":1023,"authoring":1024},1,"shape-and-space","en","extend","Projects, puzzles and the wider world of shape","Platonic solids, all 11 cube nets, rotational symmetry, tilings, olympiad problems and open questions","Build the five Platonic solids and hunt all 11 cube nets, design rangoli with rotational symmetry, explore tangram paradoxes and semi-regular tilings, count a football, see geometry in Indian monuments and nature, solve olympiad-style problems, and meet questions still unsolved.",[13,14,15,16,17],"Explain why there are exactly five Platonic solids and use duals and Euler’s formula to check their counts.","Find all 11 cube nets systematically and describe their four families.","Determine the order of rotational symmetry of shapes and designs, and explain which tilings are possible.","Solve multi-step olympiad-style problems about painted cubes, chessboards, diagonals and polygons.","Describe how shape and symmetry appear in Indian architecture, nature and careers, and state an open problem.",55,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 55 minutes plus projects",{"label":29,"value":30},"Prior knowledge","Deepen layer",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Platonic match, rotation sort, polyhedron sort, explorer",{"label":38,"value":39},"Projects","Nets, Platonic solids, rangoli, plans",[41,45,51,57,60,97,125,150,155,160,165,170,173,195,200,214,218,223,226,318,333,337,342,345,367,371,376,396,401,404,437,441,446,451,462,465,476,481,554,558,563,573,583,592,607,618,630,655,721,745,750,753,757,763,768,796,800,804,846,978,994,998,1003,1007],{"id":42,"type":43,"markdown":44},"intro-extend","prose","You now know the names, the properties, the patterns and the reasons. This last layer is about **using** all of it: projects to build with your hands, puzzles that have stumped clever people, olympiad-style problems, shapes in Indian architecture and in nature, the jobs where geometry is used every day, and questions that nobody has answered yet.\n\nPick the chapters that excite you. None of them needs to be read in order, and several are meant to take an afternoon, not ten minutes.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-extend-note","callout","observation","How to use this layer","Each chapter ends with something to **do**: build, draw, count, or argue. Work in pairs if you can; explaining a solution to someone else is the best test of whether you really understand it. Solutions are given for the problems, but try each one for at least ten minutes before peeking.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","The five perfect solids","Chapter 01","1 Platonic solids",{"id":58,"type":43,"markdown":59},"platonic","A **regular polyhedron** (or **Platonic solid**) has faces that are all the **same regular polygon**, with the **same number** of faces meeting at every vertex. The cube is one: six identical squares, three at each corner. How many others are there?\n\nThe astonishing answer, proved by Euclid more than 2,300 years ago, is: **exactly five**. Not five that people happened to find, but five that are possible, and no more, ever.",{"id":61,"type":62,"caption":63,"columns":64,"rows":71},"table-platonic","table","The five Platonic solids",[65,66,67,68,69,70],"Solid","Faces","At each vertex","F","E","V",[72,78,84,87,93],[73,74,75,76,77,76],"Tetrahedron","4 triangles","3 triangles","4","6",[79,80,81,77,82,83],"Cube (hexahedron)","6 squares","3 squares","12","8",[85,86,74,83,82,77],"Octahedron","8 triangles",[88,89,90,82,91,92],"Dodecahedron","12 pentagons","3 pentagons","30","20",[94,95,96,92,91,82],"Icosahedron","20 triangles","5 triangles",{"id":98,"type":99,"title":100,"items":101},"steps-why-five","steps","Why there are only five",[102,106,110,114,118,121],{"title":103,"tag":104,"text":105},"Corners need three","at least 3 faces","At least 3 faces must meet at each vertex; with only 2, they would fold flat and close nothing.",{"title":107,"tag":108,"text":109},"Angles must fit","total under 360°","The face angles at a vertex must add to less than 360°. At exactly 360° they lie flat; more and they cannot fit.",{"title":111,"tag":112,"text":113},"Triangles (60°)","3, 4 or 5","3 × 60 = 180, 4 × 60 = 240, 5 × 60 = 300 all work. 6 × 60 = 360 is flat. Tetrahedron, octahedron, icosahedron.",{"title":115,"tag":116,"text":117},"Squares (90°)","only 3","3 × 90 = 270 works (the cube). 4 × 90 = 360 is flat.",{"title":119,"tag":116,"text":120},"Pentagons (108°)","3 × 108 = 324 works (the dodecahedron). 4 × 108 = 432 is too much.",{"title":122,"tag":123,"text":124},"Hexagons and up","none","3 × 120 = 360 is already flat (a honeycomb). Bigger polygons are worse. So: exactly five.",{"id":126,"type":127,"component":128,"componentVersion":5,"config":129,"objective":144,"textAlternative":145,"help":146},"lab-match-platonic","interactive","match-pairs",{"prompt":130,"mode":131,"pairs":132},"Match each Platonic solid to its faces, edges and vertices.","memory",[133,135,138,140,142],{"a":73,"b":134},"4 triangles · 6 edges · 4 vertices",{"a":136,"b":137},"Cube","6 squares · 12 edges · 8 vertices",{"a":85,"b":139},"8 triangles · 12 edges · 6 vertices",{"a":88,"b":141},"12 pentagons · 30 edges · 20 vertices",{"a":94,"b":143},"20 triangles · 30 edges · 12 vertices","Remember and match the five Platonic solids with their face, edge and vertex counts.","A memory game with ten cards: five solid names and five sets of counts. The pairs are: tetrahedron with 4 triangles, 6 edges, 4 vertices; cube with 6 squares, 12 edges, 8 vertices; octahedron with 8 triangles, 12 edges, 6 vertices; dodecahedron with 12 pentagons, 30 edges, 20 vertices; icosahedron with 20 triangles, 30 edges, 12 vertices.\n\nNotice the pairs that swap: the cube and octahedron both have 12 edges, with faces and vertices swapped (6 and 8). The dodecahedron and icosahedron both have 30 edges, with 12 and 20 swapped. Every set satisfies F + V − E = 2.",{"hints":147},[148,149],"The name tells you the number of faces: tetra 4, hexa 6, octa 8, dodeca 12, icosa 20.","Pairs with the same number of edges swap faces and vertices.",{"id":151,"type":47,"variant":152,"title":153,"markdown":154},"aha-duals","aha","Solids that swap faces and vertices","Put a dot at the centre of each face of a cube and join dots on neighbouring faces. You get an **octahedron**: the cube's 6 faces become 6 vertices, and its 8 vertices become 8 faces. Do it to an octahedron and you get a cube back. These are **dual** solids. The dodecahedron and icosahedron are duals too, and the tetrahedron is its own dual. That explains the swapped numbers in the table.",{"id":156,"type":47,"variant":157,"title":158,"markdown":159},"example-dice-rpg","example","Platonic dice","Board-game shops sell dice shaped like all five Platonic solids: 4, 6, 8, 12 and 20 faces. They are fair because every face is identical and sits the same way relative to the centre, so no face is more likely than another. A dice with 7 identical, fairly placed faces is impossible in this way; that is why there is no Platonic 7-sided dice.",{"id":161,"type":47,"variant":162,"title":163,"markdown":164},"tryit-build-platonic","try_it","Build all five","Use straws and pipe cleaners, or card nets with glue tabs. Start with the tetrahedron (6 straws) and the octahedron (12 straws), which are easy and surprisingly strong. The icosahedron needs 30 straws and some patience. Check Euler's formula on each one as you finish it.",{"id":166,"type":53,"title":167,"eyebrow":168,"navLabel":169},"ch02","Hunt all 11 cube nets","Chapter 02","2 All 11 nets",{"id":171,"type":43,"markdown":172},"nets-hunt","In Investigate you learned that exactly **11** of the 35 hexominoes fold into a cube: a long-established result, not a guess. Your project: **find all 11** on squared paper, cut each one out and fold it to check. Two nets count as the same if one can be turned or flipped to match the other.\n\nThe 11 fall into four families, named by the lengths of their rows:",{"id":174,"type":62,"caption":175,"columns":176,"rows":180},"table-net-families","The four families of cube nets",[177,178,179],"Family","How many","What they look like",[181,184,188,192],[182,77,183],"1-4-1","A row of 4 squares with one square attached above and one below, anywhere along the row (the cross and the T are in this family)",[185,186,187],"1-3-2","3","A row of 3 with one square on one side, and a row of 2 on the other side overlapping it by one square",[189,190,191],"2-2-2","1","A staircase of three pairs, each pair shifted one square along",[193,190,194],"3-3","Two rows of 3 that overlap by just one square, like a long step",{"id":196,"type":47,"variant":197,"title":198,"markdown":199},"careful-count-nets","careful","Do not double count","The easiest mistake is counting the same net twice in a different orientation. Before you add a net to your collection, turn it round and flip it over (tracing paper helps) to check it does not match one you already have. A systematic hunt, starting with all the 1-4-1 nets, is the surest way to finish at exactly 11.",{"id":201,"type":62,"caption":202,"columns":203,"rows":205},"table-other-nets","How many nets do the other Platonic solids have?",[65,204],"Number of different nets",[206,208,210,211,213],[73,207],"2",[136,209],"11",[85,209],[88,212],"43,380",[94,212],{"id":215,"type":47,"variant":48,"title":216,"markdown":217},"obs-dual-nets","Duals again","The cube and octahedron are duals, and they have the same number of nets: 11. The dodecahedron and icosahedron are duals, and they share 43,380. This is not a coincidence: a net is determined by which edges you cut, and cutting patterns on dual solids correspond one to one.",{"id":219,"type":53,"title":220,"eyebrow":221,"navLabel":222},"ch03","Rotational symmetry: shapes that turn","Chapter 03","3 Rotational symmetry",{"id":224,"type":43,"markdown":225},"rot-sym","A shape has **rotational symmetry** if it looks exactly the same after being turned about its centre by some angle less than a full turn. The number of positions in one full turn where it matches itself is its **order**.\n\nThe **Ashoka Chakra** has 24 equal spokes, so it matches itself after every turn of 360° ÷ 24 = **15°**: order 24. A three-blade ceiling fan has order 3 (every 120°). A playing card like the queen of hearts, printed with the top and bottom halves upside-down copies, has order 2. Rangoli designs often have order 4 or 8.",{"id":227,"type":127,"component":228,"componentVersion":5,"config":229,"objective":312,"textAlternative":313,"help":314},"lab-sort-rotation","sort-game",{"prompt":230,"bins":231,"items":247,"seconds":311},"What is the order of rotational symmetry of each shape or object?",[232,235,238,241,244],{"id":233,"label":234},"o1","Order 1 (none)",{"id":236,"label":237},"o2","Order 2",{"id":239,"label":240},"o3","Order 3",{"id":242,"label":243},"o4","Order 4",{"id":245,"label":246},"big","Order 5 or more",[248,252,256,260,264,268,272,276,280,284,288,292,296,300,304,308],{"id":249,"label":250,"bin":245,"why":251},"r-chakra","The Ashoka Chakra (24 spokes)","It matches itself every 15°: order 24.",{"id":253,"label":254,"bin":239,"why":255},"r-fan","A three-blade ceiling fan","Every 120° the blades line up again.",{"id":257,"label":258,"bin":236,"why":259},"r-s","The letter S","A half turn (180°) brings it back to itself.",{"id":261,"label":262,"bin":233,"why":263},"r-a","The letter A","Only a full turn matches; it has a mirror line but no rotational symmetry.",{"id":265,"label":266,"bin":242,"why":267},"r-square","A square","Every quarter turn (90°).",{"id":269,"label":270,"bin":239,"why":271},"r-eq","An equilateral triangle","Every 120°.",{"id":273,"label":274,"bin":245,"why":275},"r-pent","A regular pentagon","Every 72°: order 5.",{"id":277,"label":278,"bin":236,"why":279},"r-rect","A rectangle (not a square)","Only a half turn matches; a quarter turn swaps long and short sides.",{"id":281,"label":282,"bin":242,"why":283},"r-plus","A plus sign +","Every 90°.",{"id":285,"label":286,"bin":233,"why":287},"r-kite","A kite","A kite has one mirror line but no rotational symmetry.",{"id":289,"label":290,"bin":245,"why":291},"r-hex","A honeycomb cell (regular hexagon)","Every 60°: order 6.",{"id":293,"label":294,"bin":236,"why":295},"r-para","A parallelogram (not a rectangle or rhombus)","A half turn about its centre matches it, though it has no mirror lines.",{"id":297,"label":298,"bin":245,"why":299},"r-konark","A Konark wheel (8 broad carved spokes)","The broad spokes repeat every 45°: order 8.",{"id":301,"label":302,"bin":236,"why":303},"r-z","The letter Z","Turn it upside down and it is still Z.",{"id":305,"label":306,"bin":233,"why":307},"r-iso","An isosceles (not equilateral) triangle","No turn short of 360° matches it.",{"id":309,"label":310,"bin":245,"why":275},"r-star","A five-pointed star",90,"Find the order of rotational symmetry of shapes, letters and Indian designs.","Sixteen cards and five bins, with a 90-second timer.\n\n**Order 1 (no rotational symmetry):** the letter A, a kite, an isosceles triangle.\n**Order 2:** the letters S and Z, a rectangle, a parallelogram.\n**Order 3:** a three-blade fan, an equilateral triangle.\n**Order 4:** a square, a plus sign.\n**Order 5 or more:** the Ashoka Chakra (24), a regular pentagon (5), a regular hexagon (6), a Konark wheel (8), a five-pointed star (5).\n\nThe smallest turn that works is 360° divided by the order. Some shapes, like the parallelogram and the letter S, have rotational symmetry but no line symmetry; others, like the kite and the letter A, have line symmetry but no rotational symmetry.",{"hints":315},[316,317],"Count how many times the shape matches itself during one full turn.","Turning is not reflecting: S turns onto itself but has no mirror line.",{"id":319,"type":320,"prompt":321,"options":322,"explanation":332},"pred-two-mirrors","prediction","A shape has **two** lines of symmetry that cross at right angles, like a rectangle or the letter H. Must it also have rotational symmetry?",[323,326,329],{"id":324,"label":325},"a","Yes, always: at least a half turn",{"id":327,"label":328},"b","Only if it is a square",{"id":330,"label":331},"c","No, mirror lines and turns are unrelated","**Yes, always.** Reflecting in one mirror line and then in the other is the same as turning through twice the angle between them: 2 × 90° = 180°. Since each reflection leaves the shape unchanged, so does the half turn. So any shape with two perpendicular mirror lines has rotational symmetry of order at least 2. (The reverse is false: S and Z have half-turn symmetry but no mirror lines.)",{"id":334,"type":47,"variant":162,"title":335,"markdown":336},"tryit-rangoli","Design a rangoli with order 8","On squared or dotted paper, draw a design in one 45° slice (one-eighth of a circle). Trace it and turn the tracing paper by 45° seven times, copying each time. Your finished design has rotational symmetry of order 8. Now add a mirror line through your slice's middle before copying: the result also has 8 lines of symmetry. This is how many traditional kolam and rangoli patterns are built.",{"id":338,"type":53,"title":339,"eyebrow":340,"navLabel":341},"ch04","Tangram projects and paradoxes","Chapter 04","4 Tangrams",{"id":343,"type":43,"markdown":344},"tangram-areas","Make a tangram from a square of side 8 cm (area 64 cm²). Because each piece is made by halving, the areas are simple fractions of the whole:",{"id":346,"type":62,"caption":347,"columns":348,"rows":352},"table-tangram","Tangram pieces from an 8 cm square",[349,350,351],"Piece","Area (cm²)","Fraction of the square",[353,357,360,362,364],[354,355,356],"Large triangle (each of 2)","16","1\u002F4",[358,83,359],"Medium triangle","1\u002F8",[361,83,359],"Square",[363,83,359],"Parallelogram",[365,76,366],"Small triangle (each of 2)","1\u002F16",{"id":368,"type":47,"variant":152,"title":369,"markdown":370},"aha-tangram-equal","Three pieces, three shapes, one area","The medium triangle, the square and the parallelogram look completely different, yet each has an area of exactly 8 cm²: one-eighth of the whole. You can prove it by building each one from **two small triangles**. Shapes with the same area need not look alike at all.",{"id":372,"type":47,"variant":373,"title":374,"markdown":375},"question-paradox","question","The tangram paradox","Tangram books show two figures, for example two monks, that look almost identical, except that one has a foot and the other does not. Both are made from all seven pieces. Where does the foot come from? Build both and find out. (Hint: look carefully at the body. It is slightly thinner in one of them. Area cannot appear from nowhere; it is only moved.)",{"id":377,"type":99,"title":378,"items":379},"steps-tangram-project","Tangram project ideas",[380,384,388,392],{"title":381,"tag":382,"text":383},"Alphabet","easy","Make every capital letter you can from all seven pieces. Which letters are impossible?",{"title":385,"tag":386,"text":387},"Convex shapes","medium","Exactly 13 convex shapes (no dents) can be made from all seven pieces. Find as many as you can.",{"title":389,"tag":390,"text":391},"Same pieces, new area?","think","Make a square, a rectangle and a triangle. Measure their perimeters. Same area, different perimeters: why?",{"title":393,"tag":394,"text":395},"Make your own","create","Design a new 7-piece puzzle from a rectangle. Test it on a friend.",{"id":397,"type":53,"title":398,"eyebrow":399,"navLabel":400},"ch05","Tilings: from bathrooms to jaalis","Chapter 05","5 Tilings",{"id":402,"type":43,"markdown":403},"tilings","In Investigate you found that only three regular polygons tile a floor alone: triangles, squares and hexagons. Now mix them. A **semi-regular tiling** uses two or more kinds of regular polygon, with the **same arrangement at every vertex**. We describe it by listing the polygons round a vertex: **4.8.8** means a square and two octagons (the common bathroom-floor pattern).\n\nAt each vertex the angles must add to exactly 360°. Checking every possibility, Johannes Kepler found in 1619 that there are exactly **8** semi-regular tilings:",{"id":405,"type":62,"caption":406,"columns":407,"rows":411},"table-semi","The 8 semi-regular tilings and why each fits",[408,409,410],"Vertex pattern","Angles at each vertex","Total",[412,416,419,422,425,428,431,434],[413,414,415],"3.12.12","60 + 150 + 150","360°",[417,418,415],"4.6.12","90 + 120 + 150",[420,421,415],"4.8.8","90 + 135 + 135",[423,424,415],"3.6.3.6","60 + 120 + 60 + 120",[426,427,415],"3.4.6.4","60 + 90 + 120 + 90",[429,430,415],"3.3.3.4.4","60 + 60 + 60 + 90 + 90",[432,433,415],"3.3.4.3.4","60 + 60 + 90 + 60 + 90",[435,436,415],"3.3.3.3.6","60 + 60 + 60 + 60 + 120",{"id":438,"type":47,"variant":157,"title":439,"markdown":440},"example-jaali","Jaalis and Mughal patterns","The carved stone screens (**jaalis**) of Fatehpur Sikri, the Taj Mahal and Sidi Saiyyed Mosque in Ahmedabad are masterpieces of tiling. Many use **star-and-polygon** patterns built on regular grids of hexagons, octagons and 12-gons, with 6-fold, 8-fold or 12-fold rotational symmetry repeated across the whole screen. The craftsmen planned them with a compass and straight edge, the same tools you use for constructions.",{"id":442,"type":47,"variant":443,"title":444,"markdown":445},"nuance-crystal","nuance","Why no 5-fold floor?","A pattern that repeats regularly in two directions, like wallpaper, can only have rotational symmetry of order 1, 2, 3, 4 or 6. **Never 5**, and never more than 6. This is the *crystallographic restriction*. In 1982 Dan Shechtman found crystals (**quasicrystals**) with 5-fold symmetry that never exactly repeat. Many scientists refused to believe him; he won the Nobel Prize in Chemistry in 2011. Medieval Islamic artists had already drawn similar never-repeating patterns centuries earlier.",{"id":447,"type":53,"title":448,"eyebrow":449,"navLabel":450},"ch06","The football and friends","Chapter 06","6 Football",{"id":452,"type":453,"title":454,"problem":455,"steps":456},"we-football","worked_example","Counting a football","A classic football is stitched from 12 black pentagons and 20 white hexagons, three patches meeting at every corner. How many seams (edges) and corners (vertices) does it have? Check Euler's formula.",[457,458,459,460,461],"Faces: 12 + 20 = **32**.","Edges: pentagons give 12 × 5 = 60 sides, hexagons 20 × 6 = 120 sides: 180 sides in all.","Each seam joins two patches, so E = 180 ÷ 2 = **90 seams**.","Each corner is shared by 3 patches, so V = 180 ÷ 3 = **60 corners**.","Euler: 32 + 60 − 90 = **2** ✓.",{"id":463,"type":43,"markdown":464},"fullerene","This shape is called a **truncated icosahedron**: take an icosahedron and slice off each of its 12 corners; each cut leaves a pentagon, and each triangle becomes a hexagon. In 1985 chemists discovered a molecule of **60 carbon atoms** arranged exactly at the 60 vertices of this shape. They named it **buckminsterfullerene** (or the *buckyball*), after the architect Buckminster Fuller, famous for his **geodesic domes**: huge, light, strong domes made of triangles. Geometry that a child can count on a football turned out to describe a Nobel-prize-winning molecule.",{"id":466,"type":320,"prompt":467,"options":468,"explanation":475},"pred-pentagons","Could you make a closed ball from **only hexagons**, three meeting at each corner?",[469,471,473],{"id":324,"label":470},"Yes, with enough of them",{"id":327,"label":472},"No, it would lie flat like a honeycomb",{"id":330,"label":474},"Only if they are very small","**No.** Three hexagon angles make 120° × 3 = 360°: completely flat. You need some polygons with smaller angles, like pentagons, to make the surface curve. Using Euler's formula you can prove that a ball made of pentagons and hexagons, three at each corner, must contain **exactly 12 pentagons**, however many hexagons it has. That is true of footballs, buckyballs and some virus shells.",{"id":477,"type":53,"title":478,"eyebrow":479,"navLabel":480},"ch07","Shape in Indian architecture and nature","Chapter 07","7 India and nature",{"id":482,"type":483,"title":484,"prompt":485,"options":486},"explorer-architecture","explorer","Geometry in Indian buildings","Choose a monument to see the shapes and symmetry it is built from.",[487,499,510,521,532,543],{"id":488,"label":489,"chain":490,"badge":495,"note":498},"sanchi","Sanchi Stupa",[491,492,493,494],"Circular base","Hemispherical dome","Square railing on top","Four gateways",{"text":496,"tone":497},"Hemisphere + circle","yes","The Great Stupa at Sanchi in Madhya Pradesh, begun in the 3rd century BCE, is a solid **hemisphere** (half a sphere) on a circular base, topped by a small square railing and a triple umbrella. Four carved gateways (toranas) face the four directions, giving the whole plan rotational symmetry of order 4 when seen from above.",{"id":500,"label":501,"chain":502,"badge":507,"note":509},"golgumbaz","Gol Gumbaz",[503,504,505,506],"Cube-like hall","Octagonal towers at corners","Huge dome on top","Whispering gallery",{"text":508,"tone":497},"Cube + dome","The tomb of Mohammed Adil Shah in Vijayapura (Bijapur), Karnataka, is a giant cube topped by one of the largest domes in the world: about **44 m** across on the outside, and about 38 m across inside. Work stopped in 1656 when the Sultan died, leaving the tomb unfinished. Octagonal towers of seven floors each stand at the corners. A whisper against the wall of its circular gallery can be heard clearly on the far side, because sound travels round the curved wall.",{"id":511,"label":512,"chain":513,"badge":518,"note":520},"ranikivav","Rani ki Vav",[514,515,516,517],"Long stepped trench","Seven levels down","Pillared pavilions","Deep circular well",{"text":519,"tone":497},"Symmetry + cuboids","This 11th-century stepwell in Patan, Gujarat, a UNESCO World Heritage Site since 2014, descends through seven levels of steps and pillared pavilions to a deep cylindrical well. Its long rectangular trench is symmetrical about its central line, and its walls carry hundreds of sculptures.",{"id":522,"label":523,"chain":524,"badge":529,"note":531},"jantar","Jantar Mantar",[525,526,527,528],"Giant right triangle","Quadrant arcs","Shadow falls on scale","Time and angles read",{"text":530,"tone":497},"Triangles + arcs","The Samrat Yantra at Jaipur's Jantar Mantar, built in the 1730s for Maharaja Sawai Jai Singh II, is a huge sundial: a right-angled triangular wall about **27 m** (88 ft) tall whose sloping edge points at the pole star, with two quarter-circle scales on either side. Its shadow moves about 1 mm a second, roughly 6 cm a minute, and it is said to give Jaipur local time to about two seconds.",{"id":533,"label":534,"chain":535,"badge":540,"note":542},"lotus","Lotus Temple",[536,537,538,539],"27 marble petals","In clusters of 3","Nine sides","Nine pools around",{"text":541,"tone":497},"9-fold symmetry","The Bahá'í House of Worship in Delhi, completed in 1986, is made of **27** free-standing marble-clad petals arranged in clusters of three to form **nine** sides, with nine doors. Seen from above, it has rotational symmetry of order 9. Every petal is a doubly curved concrete shell, which made the formwork a serious geometry challenge.",{"id":544,"label":545,"chain":546,"badge":551,"note":553},"basalt","St Mary’s Island",[547,548,549,550],"Molten lava cools","Rock shrinks and cracks","Cracks meet at about 120°","Mostly hexagonal columns",{"text":552,"tone":497},"Nature’s hexagons","Off the coast of Udupi, Karnataka, St Mary's Island has columns of volcanic rock, many with roughly hexagonal cross-sections. People often call them basalt, but the rock here is an acidic lava (dacite and rhyolite), which makes these columns rare in the world. As thick lava cooled and shrank, cracks formed and met at angles near 120°, the same angle as in a honeycomb, because that arrangement releases the stress most efficiently.",{"id":555,"type":47,"variant":157,"title":556,"markdown":557},"example-nature","Symmetry in living things","Look for rotational and line symmetry in nature: a starfish has 5 arms (order 5); most flowers of the lily family have 3 or 6 petals; a snowflake always has 6-fold symmetry, because of the way water molecules fit together; butterflies and human faces have (almost) one line of symmetry. Sunflower heads hide a different pattern: their seeds sit on spirals whose numbers are usually Fibonacci numbers like 34 and 55.",{"id":559,"type":53,"title":560,"eyebrow":561,"navLabel":562},"ch08","Olympiad-style problems","Chapter 08","8 Problem set",{"id":564,"type":453,"title":565,"problem":566,"steps":567},"we-painted","The painted cube","A wooden cube is painted red on the outside and then cut into 27 small equal cubes (3 × 3 × 3). How many small cubes have 3, 2, 1 and 0 red faces? What about a 10 × 10 × 10 cube?",[568,569,570,571,572],"Three red faces: only the **corner** cubes. A cube has 8 corners: **8**.","Two red faces: cubes along an edge but not at a corner. Each of the 12 edges has 3 − 2 = 1 such cube: **12**.","One red face: cubes in the middle of a face. Each of the 6 faces has (3 − 2)² = 1: **6**.","No red face: the hidden inner cube, (3 − 2)³ = **1**. Check: 8 + 12 + 6 + 1 = 27 ✓.","For side n: 8, 12(n − 2), 6(n − 2)², (n − 2)³. For n = 10: **8, 96, 384, 512**, and 8 + 96 + 384 + 512 = 1,000 ✓.",{"id":574,"type":453,"title":575,"problem":576,"steps":577},"we-chessboard","Squares on a chessboard","How many squares of all sizes are there on an 8 × 8 chessboard?",[578,579,580,581,582],"1 × 1 squares: 8 × 8 = 64.","2 × 2 squares: the top-left corner can be in 7 columns and 7 rows: 49.","In general a k × k square fits in (9 − k) × (9 − k) positions.","Total = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = **204**.","Bonus: counting all rectangles (squares included) gives 1,296, because a rectangle is fixed by choosing 2 of the 9 vertical lines and 2 of the 9 horizontal lines: 36 × 36.",{"id":584,"type":453,"title":585,"problem":586,"steps":587},"we-octagon-triangles","Triangles from an octagon’s corners","How many triangles can be made by joining three corners of a regular octagon? How many of them share **no** side with the octagon?",[588,589,590,591],"Choosing any 3 of the 8 corners gives a triangle: 8 × 7 × 6 ÷ (3 × 2 × 1) = **56**.","Triangles sharing no side need three corners, no two of them next to each other.","Fix corner 1. Its partners must come from corners 3 to 7, not adjacent to each other: (3,5), (3,6), (3,7), (4,6), (4,7), (5,7): 6 ways.","That gives 8 × 6 = 48 counts, but each triangle is counted once from each of its 3 corners: 48 ÷ 3 = **16**.",{"id":593,"type":594,"itemId":595,"prompt":596,"check":597,"hints":601,"feedback":604},"pr-diag-equal-sides","practice","shape-and-space.extend-diagonals-equal-sides","Which polygon has exactly as many diagonals as sides? Give the number of sides.",{"kind":598,"answer":599,"tolerance":600},"number",5,0,[602,603],"Solve n(n − 3) ÷ 2 = n.","Divide both sides by n.",{"correct":605,"incorrect":606},"Yes: n(n − 3) ÷ 2 = n gives n − 3 = 2, so n = 5. A pentagon has 5 sides and 5 diagonals.","Set n(n − 3) ÷ 2 = n. Dividing by n: (n − 3) ÷ 2 = 1, so n = 5: the pentagon.",{"id":608,"type":594,"itemId":609,"prompt":610,"check":611,"hints":613,"feedback":615},"pr-diag-twice","shape-and-space.extend-diagonals-twice-sides","Which polygon has exactly **twice** as many diagonals as sides? Give the number of sides.",{"kind":598,"answer":612,"tolerance":600},7,[614],"Solve n(n − 3) ÷ 2 = 2n.",{"correct":616,"incorrect":617},"Yes: n − 3 = 4, so n = 7. A heptagon has 7 sides and 14 diagonals.","n(n − 3) ÷ 2 = 2n gives n − 3 = 4, so n = 7. Check: 7 × 4 ÷ 2 = 14 = 2 × 7.",{"id":619,"type":594,"itemId":620,"prompt":621,"check":622,"hints":624,"feedback":627},"pr-painted-4","shape-and-space.extend-painted-4","A painted 4 × 4 × 4 cube is cut into 64 small cubes. How many small cubes have **exactly two** painted faces?",{"kind":598,"answer":623,"tolerance":600},24,[625,626],"These cubes lie along the edges, but not at the corners.","Each of the 12 edges has 4 − 2 cubes like this.",{"correct":628,"incorrect":629},"Yes: 12 edges × 2 = 24.","Along each of the 12 edges, the 2 middle cubes (not the corners) have exactly two painted faces: 12 × 2 = 24.",{"id":631,"type":62,"caption":632,"columns":633,"rows":636},"table-try-these","Six more to try (answers in the last column: cover it up first!)",[634,635],"Problem","Answer",[637,640,643,646,649,652],[638,639],"How many diagonals does a 15-sided polygon have?","90 (15 × 12 ÷ 2)",[641,642],"Each interior angle of a regular polygon is 162°. How many sides?","20 (exterior 18°, 360 ÷ 18)",[644,645],"A prism has 30 edges. What shape is its base, and how many faces and vertices does it have?","A decagon (3n = 30); 12 faces, 20 vertices",[647,648],"A pyramid has 11 faces. How many edges?","20 (base has 10 sides; edges = 2n)",[650,651],"A painted 6 × 6 × 6 cube is cut into unit cubes. How many have exactly one painted face?","96 (6 × 4²)",[653,654],"How many squares of all sizes are on a 5 × 5 grid?","55 (25 + 16 + 9 + 4 + 1)",{"id":656,"type":127,"component":228,"componentVersion":5,"config":657,"objective":716,"textAlternative":717,"help":718},"lab-sort-polyhedra",{"prompt":658,"bins":659,"items":666,"seconds":715},"Is it a polyhedron (all faces flat polygons) or not?",[660,663],{"id":661,"label":662},"poly","Polyhedron",{"id":664,"label":665},"not","Not a polyhedron",[667,671,675,679,683,687,691,695,699,703,707,711],{"id":668,"label":669,"bin":661,"why":670},"p-cube","A Rubik’s cube","Six flat square faces (ignoring the rounded corners).",{"id":672,"label":673,"bin":664,"why":674},"p-football","A leather football (inflated)","Its patches bulge outwards; it is close to a sphere. The truncated icosahedron is its flat-faced model.",{"id":676,"label":677,"bin":661,"why":678},"p-pyramid","The Great Pyramid of Giza","A square base and four triangular faces.",{"id":680,"label":681,"bin":664,"why":682},"p-tumbler","A steel tumbler","Its side is a curved surface.",{"id":684,"label":685,"bin":661,"why":686},"p-tent","A ridge tent","A triangular prism: two triangles and three rectangles, all flat.",{"id":688,"label":689,"bin":664,"why":690},"p-icecream","An ice-cream cone","A cone has a curved surface.",{"id":692,"label":693,"bin":661,"why":694},"p-crystal","A salt crystal","Salt crystals grow as tiny cubes with flat faces.",{"id":696,"label":697,"bin":661,"why":698},"p-d20","A 20-sided game dice","An icosahedron: 20 flat triangular faces.",{"id":700,"label":701,"bin":664,"why":702},"p-dome","The dome of Gol Gumbaz","A smooth curved dome.",{"id":704,"label":705,"bin":661,"why":706},"p-geodesic","A geodesic dome made of triangles","It looks round but every panel is a flat triangle.",{"id":708,"label":709,"bin":661,"why":710},"p-pencil","A new hexagonal pencil (unsharpened)","A hexagonal prism: 8 flat faces.",{"id":712,"label":713,"bin":664,"why":714},"p-egg","An egg","Curved all over.",45,"Decide quickly whether real objects are polyhedra (all faces flat) or have curved surfaces.","Twelve cards, two bins, and a 45-second timer.\n\n**Polyhedra:** a Rubik's cube (cube), the Great Pyramid (square pyramid), a ridge tent (triangular prism), a salt crystal (cube), a 20-sided dice (icosahedron), a geodesic dome made of flat triangles, an unsharpened hexagonal pencil (hexagonal prism).\n\n**Not polyhedra:** an inflated football (its panels bulge), a steel tumbler (cylinder), an ice-cream cone (cone), the dome of Gol Gumbaz, an egg.\n\nThe trickiest pair: a geodesic dome looks round but is built entirely from flat triangles, so it is a polyhedron; a football is modelled by a polyhedron but, when inflated, its panels curve.",{"hints":719},[720],"Look for any curved surface: one is enough to rule it out.",{"id":722,"type":127,"component":723,"componentVersion":5,"config":724,"objective":740,"textAlternative":741,"help":742},"lab-explore-all","shape-explorer",{"solids":725,"polygons":736,"modes":737},[726,727,728,729,730,731,732,733,734,735],"cube","cuboid","triangular-prism","square-pyramid","triangular-pyramid","cylinder","cone","sphere","hexagonal-prism","pentagonal-prism",[],[738,739],"explore","count","Explore all ten solids and unfold the polyhedra into nets, then race through the counting game and check Euler’s formula.","In **explore**, pick any of ten solids. Drag the solid, or use the Turn and Tilt sliders, to rotate it; hidden edges show as dashed lines. You can highlight its faces, edges or vertices, and for a polyhedron the Net button unfolds it flat. Try unfolding each prism and pyramid and compare its net with the ones in this layer.\n\nPolyhedra (F, E, V): cube and cuboid 6, 12, 8; triangular prism 5, 9, 6; pentagonal prism 7, 15, 10; hexagonal prism 8, 18, 12; square pyramid 5, 8, 5; triangular pyramid 4, 6, 4. Each gives F + V − E = 2.\n\nCurved solids: the cylinder shows 3 faces (2 flat and 1 curved), 2 edges, 0 vertices; the cone 2, 1, 1; the sphere 1, 0, 0. They have no net button, and the lab notes that Euler's formula applies to polyhedra only.\n\nIn **count**, type F, E and V for each solid shown, as fast as you can.",{"hints":743},[744],"Every prism has two identical ends; every pyramid comes to a single apex.",{"id":746,"type":53,"title":747,"eyebrow":748,"navLabel":749},"ch09","Drawing 3D on flat paper","Chapter 09","9 3D drawing",{"id":751,"type":43,"markdown":752},"isometric","How do you draw a cube on flat paper so that it looks solid? Architects and engineers use two main methods.\n\n**Isometric drawing** uses a grid of dots arranged in triangles. Vertical edges stay vertical, and the other two directions slope at 30° to the horizontal. Every edge of a cube is drawn the same length, so you can measure from the drawing. Many puzzle books and video games use this style.\n\n**Orthographic drawing** gives separate, flat views: plan, front elevation and side elevation, lined up with each other. It is less like a picture but more precise, and it is what builders actually use.\n\n**Perspective drawing**, used by artists since the Renaissance and in Mughal and Rajput miniature paintings in their own ways, makes parallel lines meet at a vanishing point so that far things look smaller, just as the eye sees them.",{"id":754,"type":47,"variant":162,"title":755,"markdown":756},"tryit-plan","Draw your home","Make an orthographic drawing of your room or home: a **plan** (top view) with walls, doors and windows, using a scale such as 1 cm for 50 cm. Then draw the front elevation of one wall. Measure with a tape if you can. Finally, try an isometric sketch of your bed or cupboard as a cuboid on triangular dot paper.",{"id":758,"type":759,"conceptId":760,"relation":761,"explanation":762},"conn-data-ext","connection","data-handling","applied_in","Surveying a class on favourite shapes, or measuring many circles to estimate π, turns geometry questions into data to collect, display and summarise.",{"id":764,"type":53,"title":765,"eyebrow":766,"navLabel":767},"ch10","Careers and open questions","Chapter 10","10 Careers and open",{"id":769,"type":62,"caption":770,"columns":771,"rows":774},"table-careers","Where people use shape and space every day",[772,773],"Work","What geometry they use",[775,778,781,784,787,790,793],[776,777],"Architect and civil engineer","Plans and elevations, triangles for rigid frames, symmetry, domes and arches",[779,780],"Packaging designer","Nets of boxes that fold with least waste and fit on a sheet of card",[782,783],"Animator and game designer","3D models built from thousands of tiny triangles (polygon meshes), rotations and views",[785,786],"Textile and jewellery designer","Tilings, rotational symmetry, repeating block-print and kolam-like patterns",[788,789],"Crystallographer and chemist","Polyhedra and symmetry of crystals and molecules such as buckyballs",[791,792],"Space engineer","Origami folds (such as the Miura fold) to pack solar panels and antennas into rockets",[794,795],"Surveyor and map maker","Top views, scale, triangles to measure land (triangulation), contour lines",{"id":797,"type":47,"variant":373,"title":798,"markdown":799},"question-open","Open questions for curious learners","Some of these are unsolved by anyone; others are open for you.\n\n- **Dürer's problem (still open):** can every convex polyhedron be cut along some of its edges and unfolded into a single flat net with no overlaps? Albrecht Dürer drew nets of solids in 1525, but the question was actually posed by G. C. Shephard in 1975. Nobody has found a counterexample, and nobody has proved it.\n- Which convex pentagons tile the plane? In 2017 Michaël Rao announced a computer-assisted proof that the 15 known types are the only ones, 99 years after the first types were found.\n- How many ways can you fold a strip of stamps? There is still no simple formula.\n- Can you design a fair dice that is **not** a Platonic solid? (Hint: think about prisms that are long and thin, and why they need rounded ends.)\n- Why do soap bubbles meet at 120° in a foam, just like honeycomb and cooling lava? What is being minimised?",{"id":801,"type":802,"prompt":803},"reflect-extend","reflection","Choose one project from this layer (the 11 nets, the five Platonic solids, a rangoli of order 8, a tangram challenge, or a plan of your home). Describe what you made, one thing that went wrong, and what geometry fact helped you fix it.",{"id":805,"type":806,"title":807,"terms":808},"glossary-extend","glossary","Words for the wider world of shape",[809,812,815,819,822,825,828,831,834,837,840,843],{"term":810,"meaning":811},"Platonic solid","A convex polyhedron whose faces are identical regular polygons, with the same number meeting at every vertex. There are exactly five.",{"term":813,"meaning":814},"Tetrahedron \u002F octahedron \u002F dodecahedron \u002F icosahedron","Platonic solids with 4, 8, 12 and 20 faces.",{"term":816,"meaning":817,"example":818},"Dual solid","The solid made by joining the face-centres of another; faces and vertices swap.","Cube and octahedron.",{"term":820,"meaning":821},"Order of rotational symmetry","The number of positions in one full turn in which a shape looks the same.",{"term":823,"meaning":824},"Semi-regular tiling","A tiling by two or more kinds of regular polygon with the same arrangement at every vertex. There are 8.",{"term":826,"meaning":827},"Vertex configuration","The list of polygons round a vertex of a tiling, such as 4.8.8.",{"term":829,"meaning":830},"Truncated icosahedron","The football shape: 12 pentagons and 20 hexagons, 90 edges, 60 vertices.",{"term":832,"meaning":833},"Geodesic dome","A dome built from many flat triangles that together approximate a sphere.",{"term":835,"meaning":836},"Isometric drawing","A drawing of a solid on a triangular grid where all three directions are drawn to the same scale.",{"term":838,"meaning":839},"Orthographic drawing","A set of flat views (plan and elevations) of a solid, lined up with each other.",{"term":841,"meaning":842},"Quasicrystal","A structure that is ordered but never exactly repeats, and can have 5-fold symmetry.",{"term":844,"meaning":845},"Jaali","A carved stone lattice screen in Indian architecture, often built on tiling patterns.",{"id":847,"type":848,"title":849,"questions":850},"quiz-extend","quiz","Puzzles and wider contexts",[851,865,878,891,904,917,929,939,952,965],{"itemId":852,"prompt":853,"options":854,"correct":330,"why":864},"shape-and-space.extend-q-platonic","Why can there be no Platonic solid made of regular hexagons?",[855,857,859,861],{"id":324,"label":856},"Hexagons are too big",{"id":327,"label":858},"Hexagons cannot be regular",{"id":330,"label":860},"Three hexagon angles already make 360°, so they lie flat",{"id":862,"label":863},"d","There is one: the honeycomb","3 × 120° = 360°: three hexagons at a vertex lie flat (a honeycomb), so they cannot fold into a corner.",{"itemId":866,"prompt":867,"options":868,"correct":324,"why":877},"shape-and-space.extend-q-dual","The octahedron has 8 faces and 6 vertices. Its dual has…",[869,871,873,875],{"id":324,"label":870},"6 faces and 8 vertices: a cube",{"id":327,"label":872},"8 faces and 6 vertices",{"id":330,"label":874},"12 faces and 12 vertices",{"id":862,"label":876},"4 faces and 4 vertices","Taking the dual swaps faces and vertices. 6 faces and 8 vertices, with the same 12 edges, is the cube.",{"itemId":879,"prompt":880,"options":881,"correct":327,"why":890},"shape-and-space.extend-q-football","How many edges (seams) does a football of 12 pentagons and 20 hexagons have?",[882,884,886,888],{"id":324,"label":883},"60",{"id":327,"label":885},"90",{"id":330,"label":887},"120",{"id":862,"label":889},"180","(12 × 5 + 20 × 6) ÷ 2 = 180 ÷ 2 = 90, because each seam joins two patches.",{"itemId":892,"prompt":893,"options":894,"correct":327,"why":903},"shape-and-space.extend-q-chakra","What is the smallest turn that maps the Ashoka Chakra onto itself?",[895,897,899,901],{"id":324,"label":896},"10°",{"id":327,"label":898},"15°",{"id":330,"label":900},"24°",{"id":862,"label":902},"30°","24 equal spokes: 360° ÷ 24 = 15°.",{"itemId":905,"prompt":906,"options":907,"correct":330,"why":916},"shape-and-space.extend-q-semi","Which set of regular polygons can meet at a vertex of a tiling?",[908,910,912,914],{"id":324,"label":909},"Two pentagons and a square",{"id":327,"label":911},"Three pentagons",{"id":330,"label":913},"Two octagons and a square",{"id":862,"label":915},"Two hexagons and a square","135° + 135° + 90° = 360°. The others give 306°, 324° and 330°.",{"itemId":918,"prompt":919,"options":920,"correct":327,"why":928},"shape-and-space.extend-q-painted","A painted 5 × 5 × 5 cube is cut into 125 small cubes. How many have no paint at all?",[921,922,924,926],{"id":324,"label":83},{"id":327,"label":923},"27",{"id":330,"label":925},"54",{"id":862,"label":927},"64","The unpainted cubes form the inner (5 − 2) × (5 − 2) × (5 − 2) = 27 block.",{"itemId":930,"prompt":931,"options":932,"correct":330,"why":938},"shape-and-space.extend-q-crystal","Which order of rotational symmetry is impossible for a regularly repeating wallpaper pattern?",[933,934,935,937],{"id":324,"label":207},{"id":327,"label":186},{"id":330,"label":936},"5",{"id":862,"label":77},"The crystallographic restriction allows only orders 1, 2, 3, 4 and 6. Five-fold symmetry appears only in non-repeating patterns such as quasicrystals.",{"itemId":940,"prompt":941,"options":942,"correct":330,"why":951},"shape-and-space.extend-q-tangram","In a tangram, which piece has the same area as the square?",[943,945,947,949],{"id":324,"label":944},"A large triangle",{"id":327,"label":946},"A small triangle",{"id":330,"label":948},"The parallelogram",{"id":862,"label":950},"None","The square, the parallelogram and the medium triangle each have one-eighth of the total area.",{"itemId":953,"prompt":954,"options":955,"correct":330,"why":964},"shape-and-space.extend-q-durer","What is Dürer’s unsolved problem about?",[956,958,960,962],{"id":324,"label":957},"Whether π ever repeats",{"id":327,"label":959},"How many Platonic solids exist",{"id":330,"label":961},"Whether every convex polyhedron can be unfolded into one non-overlapping net",{"id":862,"label":963},"Whether hexagons tile the plane","Nobody knows whether every convex polyhedron has an edge-unfolding that does not overlap. The other questions are all settled.",{"itemId":966,"prompt":967,"options":968,"correct":324,"why":977},"shape-and-space.extend-q-geodesic","A geodesic dome looks round. Why is it a polyhedron?",[969,971,973,975],{"id":324,"label":970},"It is made of flat triangles",{"id":327,"label":972},"It has a circular base",{"id":330,"label":974},"It has no edges",{"id":862,"label":976},"It is not; it is a sphere","Every panel is a flat triangle, so its surface is made entirely of polygons.",{"id":979,"type":980,"title":981,"points":982},"cheat-extend","summary","Cheat sheet",[983,984,985,986,987,988,989,990,991,992,993],"There are exactly **five Platonic solids**: tetrahedron (4, 6, 4), cube (6, 12, 8), octahedron (8, 12, 6), dodecahedron (12, 30, 20), icosahedron (20, 30, 12), as F, E, V.","Only five, because face angles at a vertex must total less than 360°: 3, 4 or 5 triangles, 3 squares, or 3 pentagons.","**Duals** swap faces and vertices: cube ↔ octahedron, dodecahedron ↔ icosahedron; the tetrahedron is self-dual.","The cube has **11 nets** in four families: 1-4-1 (6), 1-3-2 (3), 2-2-2 (1), 3-3 (1). The tetrahedron has 2; the octahedron 11.","**Rotational symmetry** of order n: the shape matches itself n times in a full turn, every 360° ÷ n. Ashoka Chakra: order 24, every 15°.","Tangram pieces are 1\u002F4, 1\u002F4, 1\u002F8, 1\u002F8, 1\u002F8, 1\u002F16, 1\u002F16 of the square.","Three regular tilings and **eight semi-regular** tilings; the angles at each vertex total 360°. Repeating patterns can only have order 1, 2, 3, 4 or 6.","Football (truncated icosahedron): 32 faces, 90 edges, 60 vertices; always exactly 12 pentagons. Same shape as the C60 buckyball.","Painted n-cube: 8 corners, 12(n − 2) edge cubes, 6(n − 2)² face cubes, (n − 2)³ hidden.","Geometry at work: architects, packaging designers, animators, crystallographers, surveyors and space engineers.","Still open: **Dürer’s problem**, whether every convex polyhedron can be unfolded into a single net.",{"id":995,"type":759,"conceptId":996,"relation":761,"explanation":997},"conn-hcf-ext","hcf-and-lcm","Tiling a rectangular floor with the largest possible equal square tiles, or stacking boxes into a cube, uses HCF and LCM.",{"id":999,"type":759,"conceptId":1000,"relation":1001,"explanation":1002},"conn-patterns-ext","patterns","related_to","Painted-cube counts, 8, 12(n − 2), 6(n − 2)², (n − 2)³, and the diagonal numbers are shape patterns with number rules.",{"id":1004,"type":759,"conceptId":1005,"relation":1001,"explanation":1006},"conn-prime-ext","prime-and-composite","Put n dots on a circle and join every k-th dot. You draw one star in a single stroke exactly when n and k are co-prime: 7 dots, every 3rd, gives a 7-pointed star; 6 dots, every 2nd, gives two separate triangles.",{"id":1008,"type":1009,"sourceIds":1010},"sources-extend","sources",[1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021],"shape-and-space-mathsisfun-platonic","shape-and-space-mathsisfun-euler","shape-and-space-wiki-net","shape-and-space-wiki-honeycomb","shape-and-space-ncert-class7","shape-and-space-ncert-class8","shape-and-space-wiki-gol-gumbaz","shape-and-space-wiki-jantar-mantar","shape-and-space-wiki-lotus-temple","shape-and-space-wiki-st-marys","shape-and-space-sahapedia-konark",[1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021],"needs_review",{"generatedBy":1025,"notes":1026},"claude-code","Draft generated locally; pending owner review.","73889b69a541c0d1848053aa4346b09b530a07d018df27c46500cfdf2f99a0d3",{"component:match-pairs@1":1029,"component:sort-game@1":1030,"logic:practice":1031,"component:shape-explorer@1":1032,"source:shape-and-space-mathsisfun-euler":1033,"source:shape-and-space-mathsisfun-platonic":1034,"source:shape-and-space-ncert-class7":1035,"source:shape-and-space-ncert-class8":1036,"source:shape-and-space-sahapedia-konark":1037,"source:shape-and-space-wiki-gol-gumbaz":1038,"source:shape-and-space-wiki-honeycomb":1039,"source:shape-and-space-wiki-jantar-mantar":1040,"source:shape-and-space-wiki-lotus-temple":1041,"source:shape-and-space-wiki-net":1042,"source:shape-and-space-wiki-st-marys":1043},"2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","a8965f19a8548e866e5fcd7f4fec4a9adac35ad54d43c9d5c416cdf3348e6198","512952430fd4d2cad9d5ad14eb51fe077bf3637c74e2c604e6458fe725b018b6","4a5b30c2c4240904f42b67c39dc2176e4a85ade7223fe6b2b7f47916eb25e6e5","20d54bcdc9a5cc5930102f965a7435de619c7bc9731d1de2ff67ae1d0307ca5c","9516472286b1156fa1275e7ea5b6574beff3acc596b8dfe8563f5c446cb7d6b6","6ee0a23cfc409e91422c539b66809bbf228580cd24778343d949f96be8766016","1c655fb4f8386989e9e702c372fa96d872ccba0e3603dce64cb08e80e373a2f0","4336a097e06982ba6b142ea63fb6f280c2335bc502b4e3e9f754af2d63e6771f","37e109029399a1432a5df500847e5e0655326424e0d432b758c345f9aa18af14","8d78c794c41f36e1fdbbdb5f4168dbb42eb9241d649f32fce9527faaf0da9f91","d6c8b2e7ef98f5aa4726bed69ebb23d330aeb205ee01d6f8d2b426062a952f6e","59957f22b8eb70f24a6166036819597fb125a6a09e19e9a0ad923c9dc586da02",{"state":1045,"reviewer":1046,"selfReview":1047,"reviewedAt":1048,"method":1049},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899599002]