garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-xi",5 "name": "Euclid's Elements, Book XI",6 "subject": "solid_geometry",7 "variant": "classical",8 "description": "Solid geometry: planes, perpendiculars, parallelepipeds, prisms. 28 definitions, 39 propositions. Depends on Books I and VI. Source: David E. Joyce.",9 "structure": {10 "books": 11,11 "definitions": 28,12 "propositions": 39,13 "foundationTypes": [14 "foundation"15 ]16 }17 },18 "metadata": {19 "created": "2026-03-18",20 "lastUpdated": "2026-03-18",21 "version": "1.0.0",22 "license": "CC BY 4.0",23 "authors": [24 "Welz, G."25 ],26 "methodology": "Programming Framework",27 "citation": "Welz, G. (2026). Euclid's Elements Book XI Dependency Graph. Programming Framework.",28 "keywords": [29 "Euclid",30 "Elements",31 "Book XI",32 "solid geometry",33 "plane",34 "parallelepiped",35 "prism"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book XI",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html",46 "notes": "Clark University"47 }48 ],49 "nodes": [50 {51 "id": "BookI",52 "type": "foundation",53 "label": "Book I — Plane geometry",54 "shortLabel": "Book I",55 "short": "Foundation",56 "book": 1,57 "colorClass": "foundation"58 },59 {60 "id": "BookVI",61 "type": "foundation",62 "label": "Book VI — Similar figures",63 "shortLabel": "Book VI",64 "short": "Foundation",65 "book": 6,66 "colorClass": "foundation"67 },68 {69 "id": "Prop1",70 "type": "proposition",71 "label": "A part of a straight line cannot be in one plane and part in another elevated",72 "shortLabel": "Prop. XI.1",73 "short": "Line part in plane",74 "book": 11,75 "number": 1,76 "colorClass": "proposition"77 },78 {79 "id": "Prop2",80 "type": "proposition",81 "label": "If two straight lines cut one another, they lie in one plane; every triangle in one plane",82 "shortLabel": "Prop. XI.2",83 "short": "Two lines cut: one plane",84 "book": 11,85 "number": 2,86 "colorClass": "proposition"87 },88 {89 "id": "Prop3",90 "type": "proposition",91 "label": "If two planes cut one another, their intersection is a straight line",92 "shortLabel": "Prop. XI.3",93 "short": "Planes cut: line",94 "book": 11,95 "number": 3,96 "colorClass": "proposition"97 },98 {99 "id": "Prop4",100 "type": "proposition",101 "label": "If line at right angles to two lines cutting at point, also perpendicular to plane through them",102 "shortLabel": "Prop. XI.4",103 "short": "Line perpendicular to plane",104 "book": 11,105 "number": 4,106 "colorClass": "proposition"107 },108 {109 "id": "Prop5",110 "type": "proposition",111 "label": "If line at right angles to three lines meeting at point, the three lie in one plane",112 "shortLabel": "Prop. XI.5",113 "short": "Three lines from point",114 "book": 11,115 "number": 5,116 "colorClass": "proposition"117 },118 {119 "id": "Prop6",120 "type": "proposition",121 "label": "If two lines at right angles to same plane, they are parallel",122 "shortLabel": "Prop. XI.6",123 "short": "Perpendicular to same plane: parallel",124 "book": 11,125 "number": 6,126 "colorClass": "proposition"127 },128 {129 "id": "Prop7",130 "type": "proposition",131 "label": "If two lines parallel, line joining points on each is in same plane",132 "shortLabel": "Prop. XI.7",133 "short": "Parallel lines: join in plane",134 "book": 11,135 "number": 7,136 "colorClass": "proposition"137 },138 {139 "id": "Prop8",140 "type": "proposition",141 "label": "If two lines parallel, one perpendicular to plane, so is the other",142 "shortLabel": "Prop. XI.8",143 "short": "Parallel: one perpendicular",144 "book": 11,145 "number": 8,146 "colorClass": "proposition"147 },148 {149 "id": "Prop9",150 "type": "proposition",151 "label": "Lines parallel to same line but not in same plane are parallel to each other",152 "shortLabel": "Prop. XI.9",153 "short": "Parallel to same: parallel",154 "book": 11,155 "number": 9,156 "colorClass": "proposition"157 },158 {159 "id": "Prop10",160 "type": "proposition",161 "label": "Two lines meeting parallel to two meeting not in same plane: contain equal angles",162 "shortLabel": "Prop. XI.10",163 "short": "Skew lines: equal angles",164 "book": 11,165 "number": 10,166 "colorClass": "proposition"167 },168 {169 "id": "Prop11",170 "type": "proposition",171 "label": "To draw line perpendicular to given plane from given elevated point",172 "shortLabel": "Prop. XI.11",173 "short": "Perpendicular from point to plane",174 "book": 11,175 "number": 11,176 "colorClass": "proposition"177 },178 {179 "id": "Prop12",180 "type": "proposition",181 "label": "To set up line at right angles to plane from given point in it",182 "shortLabel": "Prop. XI.12",183 "short": "Perpendicular from point in plane",184 "book": 11,185 "number": 12,186 "colorClass": "proposition"187 },188 {189 "id": "Prop13",190 "type": "proposition",191 "label": "From same point two lines cannot be perpendicular to same plane on same side",192 "shortLabel": "Prop. XI.13",193 "short": "One perpendicular only",194 "book": 11,195 "number": 13,196 "colorClass": "proposition"197 },198 {199 "id": "Prop14",200 "type": "proposition",201 "label": "Planes to which same line is perpendicular are parallel",202 "shortLabel": "Prop. XI.14",203 "short": "Planes perpendicular to line: parallel",204 "book": 11,205 "number": 14,206 "colorClass": "proposition"207 },208 {209 "id": "Prop15",210 "type": "proposition",211 "label": "Two lines meeting parallel to two meeting not in same plane: planes through them parallel",212 "shortLabel": "Prop. XI.15",213 "short": "Skew lines: planes parallel",214 "book": 11,215 "number": 15,216 "colorClass": "proposition"217 },218 {219 "id": "Prop16",220 "type": "proposition",221 "label": "If two parallel planes cut by any plane, intersections are parallel",222 "shortLabel": "Prop. XI.16",223 "short": "Parallel planes cut: parallel",224 "book": 11,225 "number": 16,226 "colorClass": "proposition"227 },228 {229 "id": "Prop17",230 "type": "proposition",231 "label": "If two lines cut by parallel planes, they are cut in same ratios",232 "shortLabel": "Prop. XI.17",233 "short": "Parallel planes: same ratio",234 "book": 11,235 "number": 17,236 "colorClass": "proposition"237 },238 {239 "id": "Prop18",240 "type": "proposition",241 "label": "If line perpendicular to plane, all planes through it perpendicular to that plane",242 "shortLabel": "Prop. XI.18",243 "short": "Line perpendicular: planes through it",244 "book": 11,245 "number": 18,246 "colorClass": "proposition"247 },248 {249 "id": "Prop19",250 "type": "proposition",251 "label": "If two planes cutting one another perpendicular to plane, intersection perpendicular",252 "shortLabel": "Prop. XI.19",253 "short": "Planes perpendicular: intersection",254 "book": 11,255 "number": 19,256 "colorClass": "proposition"257 },258 {259 "id": "Prop20",260 "type": "proposition",261 "label": "Solid angle by three plane angles: sum of any two greater than third",262 "shortLabel": "Prop. XI.20",263 "short": "Solid angle: plane angles",264 "book": 11,265 "number": 20,266 "colorClass": "proposition"267 },268 {269 "id": "Prop21",270 "type": "proposition",271 "label": "Any solid angle contained by plane angles summing to less than four right angles",272 "shortLabel": "Prop. XI.21",273 "short": "Solid angle: less than four right",274 "book": 11,275 "number": 21,276 "colorClass": "proposition"277 },278 {279 "id": "Prop22",280 "type": "proposition",281 "label": "Three plane angles with sum of any two greater than third, equal sides: construct triangle",282 "shortLabel": "Prop. XI.22",283 "short": "Three plane angles: construct triangle",284 "book": 11,285 "number": 22,286 "colorClass": "proposition"287 },288 {289 "id": "Prop23",290 "type": "proposition",291 "label": "To construct solid angle from three plane angles (sum of any two greater than third)",292 "shortLabel": "Prop. XI.23",293 "short": "Construct solid angle",294 "book": 11,295 "number": 23,296 "colorClass": "proposition"297 },298 {299 "id": "Prop24",300 "type": "proposition",301 "label": "If solid contained by parallel planes, opposite planes equal and parallelogrammic",302 "shortLabel": "Prop. XI.24",303 "short": "Solid by parallel planes",304 "book": 11,305 "number": 24,306 "colorClass": "proposition"307 },308 {309 "id": "Prop25",310 "type": "proposition",311 "label": "Parallelepiped cut by plane parallel to opposite: base to base as solid to solid",312 "shortLabel": "Prop. XI.25",313 "short": "Parallelepiped cut: base ratio",314 "book": 11,315 "number": 25,316 "colorClass": "proposition"317 },318 {319 "id": "Prop26",320 "type": "proposition",321 "label": "To construct solid angle equal to given on given line at given point",322 "shortLabel": "Prop. XI.26",323 "short": "Construct equal solid angle",324 "book": 11,325 "number": 26,326 "colorClass": "proposition"327 },328 {329 "id": "Prop27",330 "type": "proposition",331 "label": "To describe parallelepiped similar to given on given straight line",332 "shortLabel": "Prop. XI.27",333 "short": "Similar parallelepiped on line",334 "book": 11,335 "number": 27,336 "colorClass": "proposition"337 },338 {339 "id": "Prop28",340 "type": "proposition",341 "label": "Parallelepiped cut by plane through diagonals of opposite planes: bisected",342 "shortLabel": "Prop. XI.28",343 "short": "Parallelepiped: diagonal plane bisects",344 "book": 11,345 "number": 28,346 "colorClass": "proposition"347 },348 {349 "id": "Prop29",350 "type": "proposition",351 "label": "Parallelepipeds same base, height, ends on same lines: equal",352 "shortLabel": "Prop. XI.29",353 "short": "Same base, height, same lines: equal",354 "book": 11,355 "number": 29,356 "colorClass": "proposition"357 },358 {359 "id": "Prop30",360 "type": "proposition",361 "label": "Parallelepipeds same base, height, ends not on same lines: equal",362 "shortLabel": "Prop. XI.30",363 "short": "Same base, height, different lines: equal",364 "book": 11,365 "number": 30,366 "colorClass": "proposition"367 },368 {369 "id": "Prop31",370 "type": "proposition",371 "label": "Parallelepipeds on equal bases, same height: equal",372 "shortLabel": "Prop. XI.31",373 "short": "Equal bases, same height: equal",374 "book": 11,375 "number": 31,376 "colorClass": "proposition"377 },378 {379 "id": "Prop32",380 "type": "proposition",381 "label": "Parallelepipeds same height: to one another as bases",382 "shortLabel": "Prop. XI.32",383 "short": "Same height: as bases",384 "book": 11,385 "number": 32,386 "colorClass": "proposition"387 },388 {389 "id": "Prop33",390 "type": "proposition",391 "label": "Similar parallelepipeds: to one another in triplicate ratio of corresponding sides",392 "shortLabel": "Prop. XI.33",393 "short": "Similar: triplicate ratio",394 "book": 11,395 "number": 33,396 "colorClass": "proposition"397 },398 {399 "id": "Prop34",400 "type": "proposition",401 "label": "Equal parallelepipeds: bases reciprocally proportional to heights",402 "shortLabel": "Prop. XI.34",403 "short": "Equal: bases reciprocally proportional",404 "book": 11,405 "number": 34,406 "colorClass": "proposition"407 },408 {409 "id": "Prop35",410 "type": "proposition",411 "label": "Equal plane angles, elevated lines with equal angles: perpendiculars, joins",412 "shortLabel": "Prop. XI.35",413 "short": "Equal plane angles: elevated lines",414 "book": 11,415 "number": 35,416 "colorClass": "proposition"417 },418 {419 "id": "Prop36",420 "type": "proposition",421 "label": "Three proportional lines: parallelepiped from three equals that on mean equilateral",422 "shortLabel": "Prop. XI.36",423 "short": "Three proportional: parallelepiped",424 "book": 11,425 "number": 36,426 "colorClass": "proposition"427 },428 {429 "id": "Prop37",430 "type": "proposition",431 "label": "Four proportional: similar parallelepipeds proportional; converse",432 "shortLabel": "Prop. XI.37",433 "short": "Four proportional: parallelepipeds",434 "book": 11,435 "number": 37,436 "colorClass": "proposition"437 },438 {439 "id": "Prop38",440 "type": "proposition",441 "label": "Cube opposite sides bisected, planes through: intersection and diameter bisect each other",442 "shortLabel": "Prop. XI.38",443 "short": "Cube: bisected by planes",444 "book": 11,445 "number": 38,446 "colorClass": "proposition"447 },448 {449 "id": "Prop39",450 "type": "proposition",451 "label": "Two prisms equal height, parallelogram and triangle bases, parallelogram double: equal",452 "shortLabel": "Prop. XI.39",453 "short": "Prisms: parallelogram, triangle",454 "book": 11,455 "number": 39,456 "colorClass": "proposition"457 }458 ],459 "edges": [460 {461 "from": "BookI",462 "to": "Prop1"463 },464 {465 "from": "BookVI",466 "to": "Prop1"467 },468 {469 "from": "BookI",470 "to": "Prop2"471 },472 {473 "from": "BookVI",474 "to": "Prop2"475 },476 {477 "from": "BookI",478 "to": "Prop3"479 },480 {481 "from": "BookVI",482 "to": "Prop3"483 },484 {485 "from": "BookI",486 "to": "Prop4"487 },488 {489 "from": "BookVI",490 "to": "Prop4"491 },492 {493 "from": "BookI",494 "to": "Prop5"495 },496 {497 "from": "BookVI",498 "to": "Prop5"499 },500 {501 "from": "BookI",502 "to": "Prop6"503 },504 {505 "from": "BookVI",506 "to": "Prop6"507 },508 {509 "from": "BookI",510 "to": "Prop7"511 },512 {513 "from": "BookVI",514 "to": "Prop7"515 },516 {517 "from": "BookI",518 "to": "Prop8"519 },520 {521 "from": "BookVI",522 "to": "Prop8"523 },524 {525 "from": "BookI",526 "to": "Prop9"527 },528 {529 "from": "BookVI",530 "to": "Prop9"531 },532 {533 "from": "BookI",534 "to": "Prop10"535 },536 {537 "from": "BookVI",538 "to": "Prop10"539 },540 {541 "from": "BookI",542 "to": "Prop11"543 },544 {545 "from": "BookVI",546 "to": "Prop11"547 },548 {549 "from": "BookI",550 "to": "Prop12"551 },552 {553 "from": "BookVI",554 "to": "Prop12"555 },556 {557 "from": "BookI",558 "to": "Prop13"559 },560 {561 "from": "BookVI",562 "to": "Prop13"563 },564 {565 "from": "BookI",566 "to": "Prop14"567 },568 {569 "from": "BookVI",570 "to": "Prop14"571 },572 {573 "from": "BookI",574 "to": "Prop15"575 },576 {577 "from": "BookVI",578 "to": "Prop15"579 },580 {581 "from": "BookI",582 "to": "Prop16"583 },584 {585 "from": "BookVI",586 "to": "Prop16"587 },588 {589 "from": "BookI",590 "to": "Prop17"591 },592 {593 "from": "BookVI",594 "to": "Prop17"595 },596 {597 "from": "BookI",598 "to": "Prop18"599 },600 {601 "from": "BookVI",602 "to": "Prop18"603 },604 {605 "from": "BookI",606 "to": "Prop19"607 },608 {609 "from": "BookVI",610 "to": "Prop19"611 },612 {613 "from": "BookI",614 "to": "Prop20"615 },616 {617 "from": "BookVI",618 "to": "Prop20"619 },620 {621 "from": "BookI",622 "to": "Prop21"623 },624 {625 "from": "BookVI",626 "to": "Prop21"627 },628 {629 "from": "BookI",630 "to": "Prop22"631 },632 {633 "from": "BookVI",634 "to": "Prop22"635 },636 {637 "from": "BookI",638 "to": "Prop23"639 },640 {641 "from": "BookVI",642 "to": "Prop23"643 },644 {645 "from": "BookI",646 "to": "Prop24"647 },648 {649 "from": "BookVI",650 "to": "Prop24"651 },652 {653 "from": "BookI",654 "to": "Prop25"655 },656 {657 "from": "BookVI",658 "to": "Prop25"659 },660 {661 "from": "BookI",662 "to": "Prop26"663 },664 {665 "from": "BookVI",666 "to": "Prop26"667 },668 {669 "from": "BookI",670 "to": "Prop27"671 },672 {673 "from": "BookVI",674 "to": "Prop27"675 },676 {677 "from": "BookI",678 "to": "Prop28"679 },680 {681 "from": "BookVI",682 "to": "Prop28"683 },684 {685 "from": "BookI",686 "to": "Prop29"687 },688 {689 "from": "BookVI",690 "to": "Prop29"691 },692 {693 "from": "BookI",694 "to": "Prop30"695 },696 {697 "from": "BookVI",698 "to": "Prop30"699 },700 {701 "from": "BookI",702 "to": "Prop31"703 },704 {705 "from": "BookVI",706 "to": "Prop31"707 },708 {709 "from": "BookI",710 "to": "Prop32"711 },712 {713 "from": "BookVI",714 "to": "Prop32"715 },716 {717 "from": "BookI",718 "to": "Prop33"719 },720 {721 "from": "BookVI",722 "to": "Prop33"723 },724 {725 "from": "BookI",726 "to": "Prop34"727 },728 {729 "from": "BookVI",730 "to": "Prop34"731 },732 {733 "from": "BookI",734 "to": "Prop35"735 },736 {737 "from": "BookVI",738 "to": "Prop35"739 },740 {741 "from": "BookI",742 "to": "Prop36"743 },744 {745 "from": "BookVI",746 "to": "Prop36"747 },748 {749 "from": "BookI",750 "to": "Prop37"751 },752 {753 "from": "BookVI",754 "to": "Prop37"755 },756 {757 "from": "BookI",758 "to": "Prop38"759 },760 {761 "from": "BookVI",762 "to": "Prop38"763 },764 {765 "from": "BookI",766 "to": "Prop39"767 },768 {769 "from": "BookVI",770 "to": "Prop39"771 }772 ],773 "colorScheme": {774 "foundation": {775 "fill": "#95a5a6",776 "stroke": "#7f8c8d"777 },778 "proposition": {779 "fill": "#1abc9c",780 "stroke": "#16a085"781 }782 }783}