#include "rogueviz.h" #define PSHIFT 0 #define RVPATH HYPERPATH "rogueviz/" namespace rogueviz { #ifndef RV_ALL namespace cylon { extern void enable(); extern bool cylanim; } namespace balls { struct ball { hyperpoint at; hyperpoint vel; }; extern vector balls; extern void initialize(int); } } namespace hr { namespace bricks { extern int animation; void enable(); extern void build(bool in_pair); extern void build_stair(); struct brick { euc::coord co; color_t col; int walls; hyperpoint location; hpcshape shRotWall[6]; }; extern vector bricks; } namespace pentaroll { extern void create_pentaroll(bool); } namespace ply { extern bool animated; void enable(); extern rogueviz::objmodels::model staircase; } #endif } namespace hr { namespace dmv { transmatrix xyzscale(ld x) { transmatrix T = Id; T[0][0] = T[1][1] = T[2][2] = x; return T; } using namespace rogueviz::pres; using namespace hr::tour; struct dmv_grapher : grapher { dmv_grapher(transmatrix U) : grapher(-4, -4, 11, 11) { T = T * U; for(int x=-3; x<=10; x++) if(x) { line(p2(x,-3), p2(x,10), 0x8080FFFF); line(p2(-3,x), p2(10,x), 0x8080FFFF); } vid.linewidth *= 2; arrow(p2(0,-3), p2(0,10), .5); arrow(p2(-3,0), p2(10,0), .5); vid.linewidth /= 2; } }; void nil_screen(presmode mode, int id) { add_stat(mode, [id] { cmode |= sm::SIDE; flat_model_enabler fme; initquickqueue(); dmv_grapher g(Id); vid.linewidth *= 3; ld t = 1e-3; if(id == 2) { t = ticks / 1000.; if(t - floor(t) > .5) t = ceil(t); else t = floor(t) + 2 * (t - floor(t)); t -= floor(t/4)*4; ld t2 = 90 * degree * t; curvepoint(p2(0,0)); curvepoint(p2(5,5)); curvepoint(p2(5 + 2 * cos(t2),5 + 2 * sin(t2))); curvepoint(p2(0,0)); color_t col = cos(t2) > sin(t2) ? 0xFF808000 : 0x8080FF00; queuecurve(g.T, col | 0xFF, col | 0x20, PPR::LINE); } g.arrow(p2(5,5), p2(7,5), .3); g.arrow(p2(5,5), p2(5,7), .3); g.arrow(p2(5,5), p2(3,5), .3); g.arrow(p2(5,5), p2(5,3), .3); vid.linewidth /= 3; drawMonsterType(moEagle, nullptr, g.pos(5,5,1.5) * spin(-t * 90 * degree), 0xFF80D080, ticks / 1000., 0); queuestr(g.pos(7.5, 5, 1), 1., "E", 0); queuestr(g.pos(5, 7.5, 1), 1., "N", 0); queuestr(g.pos(2.5, 5, 1), 1., "W", 0); queuestr(g.pos(5, 2.5, 1), 1., "S", 0); quickqueue(); dialog::init(); // dialog::addTitle(id ? "Nil coordinates" : "Cartesian coordinates", forecolor, 150); poly_outline = 0xFFFFFFFF; dialog::addBreak(100); dialog::addInfo("start: (x,y,z)"); dialog::addBreak(50); if(id == 0) { dialog::addInfo("N: (x,y+d,z)"); dialog::addInfo("W: (x-d,y,z)"); dialog::addInfo("S: (x,y-d,z)"); dialog::addInfo("E: (x+d,y,z)"); } else { dialog::addInfo("N: (x,y+d,z+xd/2)", t == 1 ? 0xFFD500 : dialog::dialogcolor); dialog::addInfo("W: (x-d,y,z+yd/2)", t == 2 ? 0xFFD500 : dialog::dialogcolor); dialog::addInfo("S: (x,y-d,z-xd/2)", t == 3 ? 0xFFD500 : dialog::dialogcolor); dialog::addInfo("E: (x+d,y,z-yd/2)", t == 0 ? 0xFFD500 : dialog::dialogcolor); } dialog::addBreak(50); dialog::addInfo("U: (x,y,z-d)"); dialog::addInfo("D: (x,y,z+d)"); dialog::display(); dynamicval gg(geometry, gNil); println(hlog, eupush(point31(5, 0, 0)) * eupush(point31(0, 5, 0)) * eupush(point31(-5, 0, 0)) * eupush(point31(0, -5, 0)) * C0); return false; }); } ld geo_zero; void geodesic_screen(presmode mode, int id) { if(mode == pmStart) geo_zero = ticks; use_angledir(mode, id == 0); setCanvas(mode, '0'); if(mode == pmStart) { slide_backup(pmodel); slide_backup(pconf.clip_min); slide_backup(pconf.clip_max); slide_backup(vid.cells_drawn_limit); stop_game(), pmodel = mdHorocyclic, geometry = gCubeTiling, pconf.clip_min = -10000, pconf.clip_max = +100, start_game(); } add_stat(mode, [id] { cmode |= sm::SIDE; calcparam(); vid.cells_drawn_limit = 0; drawthemap(); // flat_model_enabler fme; initquickqueue(); dmv_grapher g(MirrorZ * cspin(1, 2, .3 * angle / (M_PI/2)) * spin(angle/2)); ld val = 25; ld rv = sqrt(val); // pi*rad*rad == 25 ld rad = sqrt(val / M_PI); ld rr = rad * sqrt(1/2.); ld radh = sqrt(val / M_PI - 2); ld rrh = radh * sqrt(1/2.); ld zmove = val - M_PI * radh * radh; ld len = hypot(2 * M_PI * radh, zmove); ld t = (ticks - geo_zero) / 500; auto frac_of = [&] (ld z) { return t - z * floor(t/z); }; t = frac_of(val); auto draw_path = [&] (auto f, color_t col) { vid.linewidth *= 5; for(ld t=0; t<=25; t+=1/16.) curvepoint(f(t)); queuecurve(g.T, col, 0, PPR::LINE); auto be_shadow = [&] (hyperpoint& h) { // ld part = 1 - angle / (M_PI / 2); // h[0] += h[2] * part / 10; h[2] = 0; }; for(ld t=0; t<=25; t+=1/16.) { hyperpoint h = f(t); be_shadow(h); curvepoint(h); } queuecurve(g.T, col & 0xFFFFFF40, 0, PPR::LINE); vid.linewidth /= 5; hyperpoint eaglepos = f(t); hyperpoint next_eaglepos = f(t + 1e-2); // queuepolyat(g.pos(x+z * .1,y,1.5) * spin(s), cgi.shEagle, 0x40, PPR::MONSTER_SHADOW).outline = 0; drawMonsterType(moEagle, nullptr, g.T * eupush(eaglepos) * rspintox(next_eaglepos - eaglepos) * xyzscale(2), col >> 8, t, 0); be_shadow(eaglepos); be_shadow(next_eaglepos); auto& bp = cgi.shEagle; println(hlog, tie(bp.shs, bp.she)); if(bp.she > bp.shs && bp.she < bp.shs + 1000) { auto& p = queuepolyat(g.T * eupush(eaglepos) * rspintox(next_eaglepos - eaglepos) * xyzscale(2), bp, 0x18, PPR::TRANSPARENT_SHADOW); p.outline = 0; p.subprio = -100; p.offset = bp.shs; p.cnt = bp.she - bp.shs; p.flags &=~ POLY_TRIANGLES; p.tinf = NULL; return; } }; color_t straight = 0x80FF80FF; color_t square = 0xcd7f32FF; color_t circle = 0xaaa9adFF; color_t helix = 0xFFD500FF; if(id >= 0) draw_path([&] (ld t) { return point31(0, 0, t); }, straight); if(id >= 1) draw_path([&] (ld t) { if(t < rv) return point31(t, 0, 0); else if(t < rv*2) return point31(rv, t-rv, rv*(t-rv)/2); else if(t < rv*3) return point31(rv-(t-rv*2), rv, rv*rv/2 + rv*(t-2*rv)/2); else if(t < rv*4) return point31(0, rv-(t-rv*3), val); else return point31(0, 0, val); }, square); if(id >= 2) draw_path([&] (ld t) { ld tx = min(t, 2 * M_PI * rad); ld ta = tx / rad - 135 * degree; ld x = rr + rad * cos(ta); ld y = rr + rad * sin(ta); ld z = rad * tx / 2 - ((rr * x) - (rr * y)) / 2; return point31(x, y, z); }, circle); if(id >= 3) draw_path([&] (ld t) { ld tx = min(t, len); ld ta = tx / len * 2 * M_PI - 135 * degree; ld x = rrh + radh * cos(ta); ld y = rrh + radh * sin(ta); ld z = radh * radh * (tx/len*2*M_PI) / 2 - ((rrh * x) - (rrh * y)) / 2 + zmove * tx / len; return point31(x, y, z); }, helix); auto cat = [] (PPR x) { if(x == PPR::MONSTER_SHADOW) return 1; else if(x == PPR::MONSTER_BODY) return 2; else return 0; }; for(int i=1; iprio) < cat(ptds[i-1]->prio)) { swap(ptds[i], ptds[i-1]); i--; } else i++; quickqueue(); dialog::init(); dialog::addTitle("from (0,0,0) to (0,0,25)", forecolor, 150); dialog::addBreak(100); dialog::addInfo("straight upwards", straight >> 8); dialog::addInfo("25", straight >> 8); if(id >= 1) { dialog::addBreak(100); dialog::addInfo("square", square >> 8); dialog::addInfo("20", square >> 8); } else dialog::addBreak(300); if(id >= 2) { dialog::addBreak(100); dialog::addInfo("circle", circle >> 8); dialog::addInfo(fts(2 * M_PI * rad), circle >> 8); } else dialog::addBreak(300); if(id >= 3) { dialog::addBreak(100); dialog::addInfo("helix", helix >> 8); dialog::addInfo(fts(len), helix >> 8); } else dialog::addBreak(300); dialog::display(); return false; }); } // i==0 - stairs // i==1 - triangle // i==2 - double triangle void brick_slide(int i, presmode mode, eGeometry geom, eModel md, int anim) { using namespace tour; setCanvas(mode, '0'); if(mode == pmStart) { set_geometry(geom); start_game(); cgi.require_shapes(); if(i == 0) bricks::build_stair(); if(i == 1) bricks::build(false); if(i == 2) bricks::build(true); bricks::enable(); tour::slide_backup(pconf.clip_min, -100.); tour::slide_backup(pconf.clip_max, +10.); tour::slide_backup(pconf.scale, i ? .2 : 2.); tour::slide_backup(mapeditor::drawplayer, false); tour::slide_backup(pconf.rotational_nil, 0.); tour::slide_backup(vid.axes3, false); bricks::animation = anim; pmodel = md; View = Id; } clearMessages(); no_other_hud(mode); } void ply_slide(tour::presmode mode, eGeometry geom, eModel md, bool anim) { using namespace tour; if(!ply::staircase.available()) { slide_error(mode, "(model not available)"); return; } if(mode == pmStartAll) { rogueviz::objmodels::prec = 10; dynamicval g(geometry, gNil); dynamicval v(variation, eVariation::pure); dynamicval s(vid.texture_step, 1); check_cgi(); cgi.require_shapes(); } setCanvas(mode, '0'); if(mode == pmStart) { set_geometry(geom); start_game(); ply::enable(); tour::slide_backup(anims::period, 40000.); tour::slide_backup(mapeditor::drawplayer, false); tour::slide_backup(pconf.rotational_nil, 0.); tour::slide_backup(ply::animated, anim); tour::slide_backup(vid.axes3, false); tour::slide_backup(no_find_player, true); tour::slide_backup(vid.texture_step, 1); tour::slide_backup(sightranges[geom], 10.); tour::slide_backup(vid.cells_drawn_limit, 50); pmodel = md; View = Id; } clearMessages(); no_other_hud(mode); } void impossible_ring_slide(tour::presmode mode) { using namespace tour; setCanvas(mode, '0'); if(mode == pmStart) { set_geometry(gCubeTiling); start_game(); tour::slide_backup(pconf.clip_min, -100.); tour::slide_backup(pconf.clip_max, +10.); tour::slide_backup(mapeditor::drawplayer, false); tour::slide_backup(vid.axes3, false); pmodel = mdHorocyclic; View = Id; } clearMessages(); no_other_hud(mode); use_angledir(mode, true); add_temporary_hook(mode, hooks_frame, 200, [] { for(int id=0; id<2; id++) { shiftmatrix T = ggmatrix(currentmap->gamestart()); println(hlog, "angle = ", angle); if(id == 1) T = T * spin(180*degree) * xpush(1.5) * cspin(0, 2, angle) * xpush(-1.5); for(ld z: {+.5, -.5}) { for(ld d=0; d<=180; d++) curvepoint(C0 + spin(d*degree) * xtangent(1) + ztangent(z)); for(ld d=180; d>=0; d--) curvepoint(C0 + spin(d*degree) * xtangent(2) + ztangent(z)); curvepoint(C0 + spin(0) * xtangent(1) + ztangent(z)); queuecurve(T, 0xFF, 0xFF8080FF, PPR::LINE); } for(ld d=0; d<180; d+=5) for(ld x: {1, 2}) { for(int i=0; i<=5; i++) curvepoint(C0 + spin((d+i)*degree) * xtangent(x) + ztangent(.5)); for(int i=5; i>=0; i--) curvepoint(C0 + spin((d+i)*degree) * xtangent(x) + ztangent(-.5)); curvepoint(C0 + spin((d+0)*degree) * xtangent(x) + ztangent(.5)); queuecurve(T, 0xFF, 0xC06060FF, PPR::LINE); } for(ld sgn: {-1, 1}) { curvepoint(C0 + xtangent(sgn * 1) + ztangent(+.5)); curvepoint(C0 + xtangent(sgn * 2) + ztangent(+.5)); curvepoint(C0 + xtangent(sgn * 2) + ztangent(-.5)); curvepoint(C0 + xtangent(sgn * 1) + ztangent(-.5)); curvepoint(C0 + xtangent(sgn * 1) + ztangent(+.5)); queuecurve(T, 0xFF, 0x804040FF, PPR::LINE); } } }); } void enable_earth() { texture::texture_aura = true; stop_game(); set_geometry(gSphere); enable_canvas(); patterns::whichCanvas = 'F'; start_game(); texture::config.configname = "textures/earth.txc"; texture::config.load(); pmodel = mdDisk; pconf.alpha = 1000; pconf.scale = 999; texture::config.color_alpha = 255; mapeditor::drawplayer = false; fullcenter(); View = spin(4 * M_PI / 5 + M_PI / 2) * View; } slide dmv_slides[] = { {"Title Page", 123, LEGAL::ANY | QUICKSKIP | NOTITLE, "", [] (presmode mode) { empty_screen(mode); show_picture(mode, "rogueviz/nil/penrose-triangle.png"); add_stat(mode, [] { gamescreen(2); dialog::init(); dialog::addBreak(400); dialog::addTitle("playing with impossibility", dialog::dialogcolor, 150); dialog::addBreak(1600); dialog::addTitle("a presentation about Nil geometry", 0xFFC000, 75); dialog::display(); return true; }); no_other_hud(mode); } }, {"Euclidean plane", 999, LEGAL::NONE | QUICKGEO, "The sum of angles of a triangle is 180 degrees.\n\n", [] (presmode mode) { if(mode == pmStartAll) enable_canvas(); setCanvas(mode, 'F'); if(mode == pmStart) { stop_game(); slide_backup(firstland, laCanvas); slide_backup(specialland, laCanvas); set_geometry(gArchimedean); arcm::current.parse("3^6"); set_variation(eVariation::pure); slide_backup(colortables['F'][0], 0xC0FFC0); slide_backup(colortables['F'][1], 0x80FF80); slide_backup(pconf.alpha, 1); slide_backup(pconf.scale, 1); start_game(); slide_backup(patterns::whichShape, '9'); slide_backup(vid.use_smart_range, 2); slide_backup(mapeditor::drawplayer, false); } add_temporary_hook(mode, hooks_frame, 200, [] { shiftmatrix T = ggmatrix(currentmap->gamestart()); vid.linewidth *= 4; shiftpoint h1 = T * xspinpush0(0, 2); shiftpoint h2 = T * xspinpush0(120*degree, 2); shiftpoint h3 = T * xspinpush0(240*degree, 2); queueline(h1, h2, 0xFF0000FF, 4); queueline(h2, h3, 0xFF0000FF, 4); queueline(h3, h1, 0xFF0000FF, 4); vid.linewidth /= 4; }); no_other_hud(mode); } }, {"spherical triangles", 999, LEGAL::NONE | QUICKGEO, "The simplest non-Euclidean geometry is the geometry on the sphere.\n\n" "Here we see a spherical triangle with three right angles.\n\n" "For creatures restricted to just this surface, they are indeed striaght lines!\n\n" , [] (presmode mode) { setCanvas(mode, '0'); if(mode == pmStart) { tour::slide_backup(mapeditor::drawplayer, false); enable_earth(); View = Id; View = spin(3 * M_PI / 5) * View; View = spin(90*degree) * View; View = cspin(2, 0, 45 * degree) * View; View = cspin(1, 2, 30 * degree) * View; playermoved = false; tour::slide_backup(vid.axes, 0); tour::slide_backup(vid.drawmousecircle, false); tour::slide_backup(draw_centerover, false); } add_temporary_hook(mode, hooks_frame, 200, [] { shiftmatrix T = ggmatrix(currentmap->gamestart()) * spin(-3 * M_PI / 5); vid.linewidth *= 4; shiftpoint h1 = T * C0; shiftpoint h2 = T * xpush0(M_PI/2); shiftpoint h3 = T * ypush0(M_PI/2); queueline(h1, h2, 0xFF0000FF, 3); queueline(h2, h3, 0xFF0000FF, 3); queueline(h3, h1, 0xFF0000FF, 3); vid.linewidth /= 4; }); if(mode == pmStop) { texture::config.tstate = texture::tsOff; } no_other_hud(mode); } }, {"hyperbolic plane", 999, LEGAL::NONE | QUICKGEO, "Hyperbolic geometry works the opposite way to spherical geometry." "In hyperbolic geometry, the sum of angles of a triangle is less than 180 degrees.\n\n", [] (presmode mode) { if(mode == pmStartAll) enable_canvas(); setCanvas(mode, 'F'); if(mode == pmStart) { stop_game(); slide_backup(firstland, laCanvas); slide_backup(specialland, laCanvas); set_geometry(gNormal); set_variation(eVariation::bitruncated); slide_backup(colortables['F'][0], 0xC0FFC0); slide_backup(colortables['F'][1], 0x80FF80); slide_backup(pconf.alpha, 1); slide_backup(pconf.scale, 1); slide_backup(rug::mouse_control_rug, true); start_game(); slide_backup(patterns::whichShape, '9'); } if(mode == pmStart) { rug::modelscale = 1; // rug::rug_perspective = false; rug::gwhere = gCubeTiling; rug::texturesize = 1024; slide_backup(sightrange_bonus, -1); drawthemap(); rug::init(); } if(mode == pmStart) { stop_game(); set_geometry(gArchimedean); arcm::current.parse("3^7"); set_variation(eVariation::pure); start_game(); } add_temporary_hook(mode, hooks_frame, 200, [] { shiftmatrix T = ggmatrix(currentmap->gamestart()); vid.linewidth *= 16; shiftpoint h1 = T * xspinpush0(0, 2); shiftpoint h2 = T * xspinpush0(120*degree, 2); shiftpoint h3 = T * xspinpush0(240*degree, 2); queueline(h1, h2, 0xFF0000FF, 4); queueline(h2, h3, 0xFF0000FF, 4); queueline(h3, h1, 0xFF0000FF, 4); vid.linewidth /= 16; }); if(mode == 3) { rug::close(); } no_other_hud(mode); } }, {"A right-angled pentagon", 999, LEGAL::NONE | QUICKGEO, "There is also three-dimensional hyperbolic geometry.\n" "Here are some right-angled pentagons in three-dimensional hyperbolic space.\n" , [] (presmode mode) { if(mode == pmStart) { slide_backup(patterns::rwalls, 10); slide_backup(vid.fov, 120); } setCanvas(mode, '0'); if(mode == pmStart) { set_geometry(gSpace534); /* static bool solved = false; if(!solved) { stop_game(); set_geometry(gSpace534); start_game(); stop_game(); cgi.require_basics(); fieldpattern::field_from_current(); set_geometry(gFieldQuotient); currfp.Prime = 5; currfp.force_hash = 0x72414D0C; currfp.solve(); solved = true; } set_geometry(gFieldQuotient); */ start_game(); tour::slide_backup(mapeditor::drawplayer, false); pentaroll::create_pentaroll(true); tour::slide_backup(anims::period, 30000.); tour::slide_backup(sightranges[geometry], 4); start_game(); playermoved = false; } no_other_hud(mode); } }, {"Penrose triangle (1958), Oscar Reutersvärd's triangle (1934), Penrose Stairs (1959)", 999, LEGAL::NONE, "The Penrose Triangle, " "constructed by Lionel Penrose and Roger Penrose in 1958, " "is an example of an impossible figure. " "Many artists have used the Penrose Triangle to create " "more complex constructions, such as the \"Waterfall\" " "by M. C. Escher.\n\n" "While it is known as Penrose Triangle, a very similar construction " "has actually been discovered earlier by Oscar Reutersvärd (in 1934)!\n\n" "In 1959 Lionel Penrose and Roger Penrose have constructed another " "example of an impossible figure, called the Penrose staircase.", [] (presmode mode) { empty_screen(mode); show_picture(mode, "rogueviz/nil/penrose-all-small.png"); no_other_hud(mode); } }, {"Ascending & Descending", 999, LEGAL::NONE | QUICKGEO, "It is the most well known from \"Ascending and Descending\" by M. C. Escher.\n\n" "This is a 3D model of Ascending and Descending by Lucian B. It is based on an optical illusion." , [] (presmode mode) { slide_url(mode, 'm', "link to the original model", "https://3dwarehouse.sketchup.com/model/3e6df6c24a95f583cefabc2ae69d584c/MC-Escher-Ascending-and-Descending"); ply_slide(mode, gCubeTiling, mdPerspective, false); if(!ply::staircase.available()) return; if(mode == pmStart) { tour::slide_backup(smooth_scrolling, true); tour::slide_backup(sightranges[geometry], 200); tour::slide_backup(vid.cells_drawn_limit, 200); tour::slide_backup(camera_speed, 5); centerover = euc::get_at(euc::coord{12,-23,8})->c7; playermoved = false; int cid = 0; for(ld val: {0.962503,0.254657,-0.0934754,0.000555891,0.0829357,-0.604328,-0.792408,0.0992114,-0.258282,0.754942,-0.602787,0.0957558,0.,0.,0.,1.}) View[0][cid++] = val; // tour::slide_backup(vid.fov, 120); } if(mode == pmKey) { println(hlog, ggmatrix(currentmap->gamestart())); println(hlog, View); println(hlog, euc::get_ispacemap()[centerover->master]); } } }, /* {"Penrose triangle (1958)", 999, LEGAL::NONE, "The Penrose Triangle, " "constructed by Lionel Penrose and Roger Penrose in 1958, " "is an example of an impossible figure. " "Many artists have used the Penrose Triangle to create " "more complex constructions, such as the \"Waterfall\" " "by M. C. Escher.", [] (presmode mode) { empty_screen(mode); show_picture(mode, "rogueviz/nil/penrose-triangle.png"); no_other_hud(mode); } }, {"Oscar Reutersvärd's triangle (1934)", 999, LEGAL::NONE, "While it is known as Penrose Triangle, a very similar construction " "has actually been discovered earlier by Oscar Reutersvärd (in 1934)!", [] (presmode mode) { empty_screen(mode); show_picture(mode, "rogueviz/nil/reutersvard.png"); no_other_hud(mode); } }, {"Penrose staircase (1959)", 999, LEGAL::NONE, "In 1959 Lionel Penrose and Roger Penrose have constructed another " "example of an impossible figure, called the Penrose staircase.\n\n" "It is the most well known from \"Ascending and Descending\" by M. C. Escher.\n\n", [] (presmode mode) { empty_screen(mode); show_picture(mode, "rogueviz/nil/penrose-stairs.png"); no_other_hud(mode); } }, */ {"non-Euclidean geometry so far", 123, LEGAL::ANY, "People sometimes call such impossible constructions \"non-Euclidean\".\n\n" "These people generally use this name because they do not know the usual " "mathematical meaning of \"non-Euclidean\".\n\n" "It seems that the geometries we know so far are something completely different...", [] (presmode mode) { empty_screen(mode); add_stat(mode, [] { dialog::init(); color_t d = dialog::dialogcolor; dialog::addTitle("Euclidean geometry", 0xC00000, 200); dialog::addTitle("lines stay parallel", d, 150); dialog::addBreak(100); dialog::addTitle("spherical geometry", 0xC00000, 200); dialog::addTitle("lines converge", d, 150); dialog::addBreak(100); dialog::addTitle("hyperbolic geometry", 0xC00000, 200); dialog::addTitle("lines diverge", d, 150); dialog::display(); return true; }); no_other_hud(mode); } }, {"a Puzzle about a Bear", 123, LEGAL::ANY, "However, it turns out that there actually exists a non-Euclidean geometry, " "known as the Nil geometry, where constructions such as Penrose staircases and " "triangles naturally appear!\n\n" "Nil is a three-dimensional geometry, which gives new possibilities -- " "Lines 'diverge in the third dimension' there. " "To explain Nil geometry, we will start with a well-known puzzle.", [] (presmode mode) { empty_screen(mode); add_stat(mode, [] { dialog::init(); color_t d = dialog::dialogcolor; dialog::addTitle("A bear walked five kilometers north, ", d, 150); dialog::addTitle("five kilometers east, five kilometers south, ", d, 150); dialog::addTitle("and returned exactly to the place it started.", d, 150); dialog::addBreak(50); dialog::addTitle("What color is the bear?", 0xC00000, 200); dialog::display(); return true; }); no_other_hud(mode); } }, {"Cartesian coordinates", 999, LEGAL::NONE, "The puzzle shows an important fact: every point on Earth has defined directions " "(North, East, South, West), and in most life situations, we can assume that these " "directions work the same as in the Cartesian system of coordinates." , [] (presmode mode) { empty_screen(mode); nil_screen(mode, 0); no_other_hud(mode); } }, {"Nil coordinates", 999, LEGAL::NONE, "However, because Earth is curved (non-Euclidean), these directions actually " "work different! If you are closer to the pole, moving East or West changes " "your longitude much more quickly.\n\n" "Nil is a three-dimensional geometry which is similar: while every point also has " "well-defined NSEWUD directions, but they affect the coordinates in a different way " "than in the Euclidean space with the usual coordinate system.\n\n" "You may want to use the Pythagorean theorem to compute the length of these -- " "this is not correct, all the moves are of length d. You would need to use the Pythagorean " "theorem if you wanted to compute the length from (x,y,z) to (x,y-d,z).\n\n" , [] (presmode mode) { empty_screen(mode); nil_screen(mode, 1); no_other_hud(mode); } }, {"Nil coordinates (area)", 999, LEGAL::NONE, "The formulas look strange at a first glance, but the idea is actually simple: " "the change in the 'z' coordinate is the area of a triangle, as shown in the picture. " "The change is positive if we go counterclockwise, and negative if we go clockwise.\n\n" "If we make a tour in Nil moving only in the directions N, E, S, W, such that " "the analogous tour in Euclidean space would return us to the starting point, " "then the tour in Nil would return us directly above or below the starting point, " "with the difference in the z-coordinate proportional to the area of the loop." , [] (presmode mode) { empty_screen(mode); nil_screen(mode, 2); no_other_hud(mode); } }, {"Simple Penrose stairs", 999, LEGAL::NONE | QUICKGEO, "This lets us easily make a simple realization of the Penrose staircase in Nil. " "Here is an attempt to create a Penrose staircase in Euclidean geometry...\n\n" "(you can rotate this with mouse or arrow keys)" , [] (presmode mode) { brick_slide(0, mode, gCubeTiling, mdHorocyclic, 0); } }, {"Simple Penrose stairs in Nil", 999, LEGAL::NONE | QUICKGEO, "We can use the magic of the Nil geometry to recompensate the lost height.\n\n" "Press 5 to see how it looks when we walk around the stairs. When you rotate this slide, " "you will notice that the stairs change shape when far from the central point -- " "this is because we use the Nil rules of movement." , [] (presmode mode) { brick_slide(0, mode, gNil, mdHorocyclic, 0); if(mode == pmKey) bricks::animation = !bricks::animation; } }, {"Simple Penrose stairs in Nil (FPP)", 999, LEGAL::NONE | QUICKGEO, "This slide shows our stairs in the first person perspective, from the inside." , [] (presmode mode) { brick_slide(0, mode, gNil, mdPerspective, 3); if(mode == pmKey) bricks::animation ^= 1; } }, {"Geodesics in Nil", 999, LEGAL::NONE | QUICKGEO, "But, was the first person perspective in the last slide 'correct'?\n\n" "According to Fermat's Principle, the path taken by a light ray is " "always one which is the shortest. Our previous visualization assumed " "that light rays move in a fixed 'direction', which may be not the case.\n\n" "Let's think a bit about moving from (0,0,0) to (0,0,25). We can of course " "take the obvious path of length 25. Can we do it better?" , [] (presmode mode) { empty_screen(mode); geodesic_screen(mode, 0); no_other_hud(mode); } }, {"Geodesics: square", 999, LEGAL::NONE | QUICKGEO, "Yes, we can! Here is a square of edge length 5. Since such a square has an " "area of 25 and perimeter of 20, it takes us to (0,0,25) in just 20 steps!" , [] (presmode mode) { empty_screen(mode); geodesic_screen(mode, 1); no_other_hud(mode); } }, {"Geodesics: circle", 999, LEGAL::NONE | QUICKGEO, "We can do even better. Queen Dido already knew that among shapes with the " "given area, the circle has the shortest perimeter. A circle with area 25 " "has even shorter length." , [] (presmode mode) { empty_screen(mode); geodesic_screen(mode, 2); no_other_hud(mode); } }, {"Geodesics: helix", 999, LEGAL::NONE | QUICKGEO, "But that was just the silver medal.\n\n" "For the gold medal, we need to combine the 'silver' and 'green' paths. " "We make the circle slightly smaller, and we satisfy the difference by moving " "slightly upwards. The length of such path can be computed using the Pythagorean " "theorem, and minimized by differentiation. There is an optimal radius which " "yields the best path.", [] (presmode mode) { empty_screen(mode); geodesic_screen(mode, 3); no_other_hud(mode); } }, {"Simple Penrose stairs in Nil (geodesics)", 999, LEGAL::NONE | QUICKGEO, "The light ray paths ('geodesics') in Nil are like the ones constructed in " "the last slide: they are helices, the steeper the helix, the smaller " "its radius.\n\n" "This slide presents the staircase in model perspective and the " "geodesically correct view. The geodesically correct view appears to spin." , [] (presmode mode) { brick_slide(0, mode, gNil, mdGeodesic, 3); compare_projections(mode, mdPerspective, mdGeodesic); } }, {"Penrose triangle (illusion)", 999, LEGAL::NONE | QUICKGEO, "Can we also construct the Penrose triangle? " "Yes, we can! In our space, we can construct an illusion " "which looks like the Penrose triangle (rotate the scene and press '5'). " "If we rotate this illusion in such a way that the 'paradox line' " "is vertical, we can recompensate the difference by using the Nil geometry. " "We need to scale our scene in such a way that the length of the white line " "equals the area contained in the projection of the red line." , [] (presmode mode) { brick_slide(1, mode, gCubeTiling, mdHorocyclic, 0); static bool draw = false; if(mode == pmKey) draw = !draw; add_temporary_hook(mode, hooks_prestats, 200, [] { if(draw) { shiftmatrix Zero = ggmatrix(currentmap->gamestart()); initquickqueue(); // two first bricks are fake int id = 0; for(auto& b: bricks::bricks) { id++; if(id >= 2) curvepoint(b.location); } vid.linewidth *= 10; queuecurve(Zero, 0x0000FFFF, 0, PPR::SUPERLINE).flags |= POLY_FORCEWIDE; vid.linewidth /= 10; curvepoint(bricks::bricks[2].location); curvepoint(bricks::bricks.back().location); vid.linewidth *= 10; queuecurve(Zero, 0xFFFFFFFF, 0, PPR::SUPERLINE).flags |= POLY_FORCEWIDE; vid.linewidth /= 10; quickqueue(); } return false; }); // pmodel = (pmodel == mdGeodesic ? mdPerspective : mdGeodesic); } }, {"Penrose triangle (Nil)", 999, LEGAL::NONE | QUICKGEO, "Here we move around the Penrose triangle..." , [] (presmode mode) { brick_slide(1, mode, gNil, mdHorocyclic, 1); // if(mode == pmKey) DRAW // pmodel = (pmodel == mdGeodesic ? mdPerspective : mdGeodesic); } }, {"Penrose triangle (FPP)", 999, LEGAL::NONE | QUICKGEO, "... and see the Penrose triangle in first-person perspective. " "Since the Penrose triangle is larger (we need stronger Nil effects " "to make it work), the geodesic effects are also much stronger." , [] (presmode mode) { brick_slide(1, mode, gNil, mdPerspective, 3); compare_projections(mode, mdPerspective, mdGeodesic); } }, {"Improbawall by Matt Taylor (emty01)", 999, LEGAL::NONE | QUICKGEO, "This impossible construction by Matt Taylor was popular in early 2020. " "How does it even work?\n\n" "(the animation is not included with RogueViz)" , [] (presmode mode) { static bool pic_exists, video_exists; if(mode == pmStartAll || mode == pmStart) { pic_exists = file_exists("rogueviz/nil/emty-ring.png"); video_exists = file_exists("rogueviz/nil/emty-ring.mp4"); } slide_url(mode, 'i', "Instagram link", "https://www.instagram.com/p/B756GCynErw/"); empty_screen(mode); if(video_exists) show_animation(mode, "rogueviz/nil/emty-ring.mp4", 720, 900, 300, 30); else if(pic_exists) show_picture(mode, "rogueviz/nil/emty-ring.png"); else slide_error(mode, "(image not available)"); no_other_hud(mode); } }, {"how is this made", 999, LEGAL::NONE | QUICKGEO, "Rotate this ring and press '5' to rotate a half of it by 90 degrees. " "After rotating this ring so that the endpoints agree, we get another " "case that can be solved in Nil geometry." , [] (presmode mode) { impossible_ring_slide(mode); } }, {"impossible ring in Nil", 18, LEGAL::NONE | QUICKGEO, "Here is how it looks in Nil. Press '5' to animate.\n", [] (presmode mode) { setCanvas(mode, '0'); slidecommand = "animation"; if(mode == pmKey) { tour::slide_backup(rogueviz::cylon::cylanim, !rogueviz::cylon::cylanim); } if(mode == pmStart) { stop_game(); set_geometry(gNil); tour::slide_backup(mapeditor::drawplayer, false); rogueviz::cylon::enable(); tour::slide_backup(smooth_scrolling, true); tour::on_restore(nilv::set_flags); tour::slide_backup(nilv::nilperiod, make_array(3, 3, 3)); nilv::set_flags(); start_game(); playermoved = false; } }}, {"3D model (geodesic)", 999, LEGAL::NONE | QUICKGEO, "What if we try to move something more complex, rather than a simple geometric shape?\n\n" "This slide is based on a 3D model of Ascending and Descending by Lucian B. " "We have used the trick mentioned before to move into the Nil space. Here are the results." , [] (presmode mode) { slide_url(mode, 'y', "YouTube link", "https://www.youtube.com/watch?v=DurXAhFrmkE"); ply_slide(mode, gNil, mdGeodesic, true); } }, {"3D model (perspective)", 999, LEGAL::NONE | QUICKGEO, "The same in the simple model." , [] (presmode mode) { ply_slide(mode, gNil, mdPerspective, true); } }, {"two Penrose triangles (Euc)", 999, LEGAL::NONE | QUICKGEO, "Here are two Penrose triangles. Can we move that to Nil?" , [] (presmode mode) { brick_slide(2, mode, gCubeTiling, mdHorocyclic, 0); } }, {"two Penrose triangles (Nil)", 999, LEGAL::NONE | QUICKGEO, "No, we cannot -- one of the triangles has opposite orientation!\n\n" "That is still impossible in Nil, so not all " "impossible constructions can be realized in Nil.\n\n" "For example, \"Waterfall\" by M. C. Escher is based on three " "triangles with two different orientations.", [] (presmode mode) { brick_slide(2, mode, gNil, mdHorocyclic, 0); } }, {"Balls in Nil", 999, LEGAL::NONE | QUICKGEO | FINALSLIDE, "A perpetuum mobile in Nil as the final slide. That's all for today!" , [] (presmode mode) { slide_url(mode, 'y', "YouTube link", "https://www.youtube.com/watch?v=mxvUAcgN3go"); setCanvas(mode, '0'); if(mode == pmStart) { stop_game(); set_geometry(gNil); check_cgi(); cgi.require_shapes(); tour::slide_backup(mapeditor::drawplayer, false); tour::slide_backup(smooth_scrolling, true); start_game(); rogueviz::balls::initialize(1); rogueviz::balls::balls.resize(3); pmodel = mdEquidistant; playermoved = false; View = cspin(1, 2, M_PI/2); } } }, {"final slide", 123, LEGAL::ANY | NOTITLE | QUICKSKIP | FINALSLIDE, "FINAL SLIDE", [] (presmode mode) { empty_screen(mode); add_stat(mode, [] { dialog::init(); color_t d = dialog::dialogcolor; dialog::addTitle("Thanks for your attention!", 0xC00000, 200); dialog::addBreak(100); dialog::addTitle("twitter.com/zenorogue/", d, 150); dialog::display(); return true; }); no_other_hud(mode); } } }; int phooks = 0 + addHook_slideshows(100, [] (tour::ss::slideshow_callback cb) { cb(XLAT("Playing with Impossibility"), &dmv_slides[0], 'i'); }); } } // kolor zmienic dziada