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Christmas Card 3: Geometric Symmetry and Arithmetic Mean in Festive Design

Christmas Card 3: Geometric Symmetry and Arithmetic Mean in Festive Design
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2021-07-09 21:30
CirclePolygonSequenceTransformationGeometric Construction
In collections STEAM like Leonardo
This example demonstrates a Christmas card themed geometric pattern, constructed using circles, points, segments, and sequences, combined with symmetry and arithmetic mean concepts to create a decorative festive design. The core mathematical concepts include properties of quadrilaterals, arithmetic mean (used for midpoints or symmetric positions), perimeter calculation, arithmetic operations, and symmetry transformations. The construction uses GeoGebra commands such as Circle, Point, Segment, and Sequence to create a series of points (A through K), segments (l1 to l16), and a circle (c), with symmetric operations and sequences generating repeated decorative elements, finally embedding text and an image (obraz1) to complete the card. This is suitable for middle school students learning geometric construction, symmetry transformations, and applications of arithmetic mean, and can also be used in interdisciplinary activities combining mathematics and art to inspire interest in geometric beauty.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. Point(c) → E
  5. Point(c) → F
  6. Point(c) → G
  7. Point(c) → H
  8. Point(c) → I
  9. Segment(A, B) → f
  10. Sequence(A + (i / n * (B - A)), i, 1, n) → l1, {(-6.3014, -2.452913432835821), (-6.3014, -2.278226865671642), (-6.3014, -2.103540298507463), (-6.3014, -1.9288537313432839), (-6.3014, -1.7541671641791048), (-6.3014, -1.5794805970149255), (-6.3014, -1.4047940298507464), (-6.3014, -1.2301074626865673), (-6.3014, -1.055420895522388), (-6.3014, -0.8807343283582092), (-6.3014, -0.7060477611940299), (-6.3014, -0.5313611940298508), (-6.3014, -0.35667462686567175), (-6.3014, -0.18198805970149268), (-6.3014, -0.007301492537313159), (-6.3014, 0.16738507462686547), (-6.3014, 0.342071641791045), (-6.3014, 0.5167582089552241), (-6.3014, 0.6914447761194027), (-6.3014, 0.8661313432835818), (-6.3014, 1.0408179104477613), (-6.3014, 1.2155044776119404), (-6.3014, 1.3901910447761194), (-6.3014, 1.5648776119402985), (-6.3014, 1.7395641791044776), (-6.3014, 1.9142507462686567), (-6.3014, 2.0889373134328357), (-6.3014, 2.263623880597015), (-6.3014, 2.438310447761194), (-6.3014, 2.612997014925374), (-6.3014, 2.787683582089552), (-6.3014, 2.962370149253731), (-6.3014, 3.13705671641791)}
  11. Segment(A, C) → g
  12. Sequence(A + (i / n * (C - A)), i, 1, n) → l2, {(-6.175521715248428, -2.5064800009263233), (-6.049643430496857, -2.385360001852646), (-5.923765145745286, -2.264240002778969), (-5.797886860993714, -2.1431200037052918), (-5.672008576242143, -2.022000004631615), (-5.546130291490571, -1.9008800055579376), (-5.420252006738999, -1.7797600064842607), (-5.294373721987428, -1.6586400074105834), (-5.1684954372358565, -1.5375200083369065), (-5.042617152484285, -1.4164000092632294), (-4.916738867732713, -1.2952800101895523), (-4.790860582981142, -1.1741600111158752), (-4.66498229822957, -1.053040012042198), (-4.539104013477999, -0.931920012968521), (-4.413225728726427, -0.8108000138948439), (-4.287347443974856, -0.6896800148211668), (-4.161469159223284, -0.5685600157474897), (-4.035590874471713, -0.44744001667381283), (-3.9097125897201415, -0.3263200176001355), (-3.78383430496857, -0.20520001852645864), (-3.657956020216998, -0.08408001945278132), (-3.5320777354654265, 0.03703997962089556), (-3.406199450713855, 0.15815997869457243), (-3.2803211659622837, 0.27927997776824975), (-3.154442881210712, 0.4003999768419271), (-3.0285645964591406, 0.521519975915604), (-2.902686311707569, 0.6426399749892808), (-2.776808026955998, 0.7637599740629581), (-2.6509297422044265, 0.884879973136635), (-2.525051457452854, 1.0059999722103123), (-2.399173172701283, 1.1271199712839892), (-2.273294887949712, 1.2482399703576665), (-2.14741660319814, 1.3693599694313434)}
  13. Sequence(Segment[Element[l1, i], Element[l2, n - i]], i, 1, n) → l3, {5.644940391411102, 5.353652878114178, 5.066018159313959, 4.782695325447797, 4.50449807370638, 4.232437235257201, 3.9677752883904116, 3.7120951310997494, 3.4673842475142806, 3.2361320294278384, 3.0214303298719565, 2.827052475468831, 2.6574619271480797, 2.5176726999019508, 2.4128699691082476, 2.347743786323403, 2.3256298112594624, 2.3477437863234036, 2.412869969108247, 2.5176726999019508, 2.6574619271480797, 2.8270524754688307, 3.021430329871957, 3.2361320294278384, 3.46738424751428, 3.712095131099749, 3.967775288390411, 4.2324372352572, 4.50449807370638, 4.782695325447798, 5.066018159313958, 5.353652878114178, 5.644940391411101}
  14. Segment(A, D) → h
  15. Sequence(A + (i / n * (D - A)), i, 1, n) → l4, {(-6.126718847878435, -2.6262245578573107), (-5.95203769575687, -2.624849115714621), (-5.777356543635305, -2.6234736735719317), (-5.60267539151374, -2.6220982314292423), (-5.4279942393921745, -2.620722789286553), (-5.25331308727061, -2.6193473471438633), (-5.078631935149044, -2.617971905001174), (-4.903950783027479, -2.6165964628584844), (-4.729269630905915, -2.615221020715795), (-4.554588478784349, -2.6138455785731054), (-4.379907326662784, -2.612470136430416), (-4.2052261745412185, -2.6110946942877264), (-4.030545022419654, -2.609719252145037), (-3.8558638702980885, -2.6083438100023475), (-3.6811827181765233, -2.606968367859658), (-3.5065015660549586, -2.6055929257169685), (-3.3318204139333933, -2.604217483574279), (-3.157139261811828, -2.6028420414315896), (-2.9824581096902634, -2.6014665992889), (-2.807776957568698, -2.6000911571462106), (-2.6330958054471325, -2.598715715003521), (-2.4584146533255677, -2.5973402728608317), (-2.2837335012040025, -2.595964830718142), (-2.1090523490824378, -2.5945893885754527), (-1.9343711969608721, -2.5932139464327633), (-1.7596900448393074, -2.591838504290074), (-1.5850088927177426, -2.5904630621473843), (-1.410327740596177, -2.589087620004695), (-1.2356465884746122, -2.5877121778620054), (-1.0609654363530465, -2.586336735719316), (-0.8862842842314818, -2.5849612935766264), (-0.711603132109917, -2.583585851433937), (-0.5369219799883513, -2.582210409291248)}
  16. Sequence(Segment[Element[l2, i], Element[l4, n - i]], i, 1, n) → l5, {5.639108268975346, 5.341719546610919, 5.047680217336353, 4.757611339186605, 4.472285539793912, 4.192671279812797, 3.919990967860639, 3.6557965113057787, 3.402065445059307, 3.1613181853907464, 2.936749608414459, 2.732351747522141, 2.5529739713276935, 2.4042230222653367, 2.29206954513366, 2.2220619872299587, 2.1982307355871593, 2.222061987229959, 2.29206954513366, 2.4042230222653367, 2.5529739713276935, 2.7323517475221415, 2.936749608414459, 3.161318185390747, 3.402065445059308, 3.6557965113057787, 3.9199909678606404, 4.192671279812798, 4.472285539793912, 4.757611339186606, 5.047680217336353, 5.341719546610919, 5.639108268975347}
  17. Segment(A, E) → i
  18. Sequence(A + (i / n * (E - A)), i, 1, n) → l6, {(-6.174489544446484, -2.747638048213074), (-6.047579088892967, -2.867676096426147), (-5.92066863333945, -2.9877141446392206), (-5.793758177785934, -3.1077521928522938), (-5.666847722232417, -3.2277902410653674), (-5.5399372666789, -3.347828289278441), (-5.413026811125383, -3.467866337491514), (-5.286116355571867, -3.5879043857045874), (-5.1592059000183506, -3.707942433917661), (-5.032295444464834, -3.8279804821307346), (-4.905384988911317, -3.9480185303438082), (-4.7784745333578, -4.068056578556881), (-4.6515640778042835, -4.188094626769955), (-4.524653622250767, -4.308132674983028), (-4.39774316669725, -4.428170723196102), (-4.270832711143734, -4.5482087714091755), (-4.143922255590217, -4.668246819622249), (-4.0170118000367, -4.788284867835322), (-3.890101344483184, -4.908322916048395), (-3.7631908889296675, -5.028360964261468), (-3.6362804333761503, -5.148399012474543), (-3.509369977822634, -5.268437060687615), (-3.382459522269117, -5.388475108900689), (-3.2555490667156004, -5.508513157113763), (-3.1286386111620836, -5.628551205326836), (-3.001728155608567, -5.74858925353991), (-2.8748177000550506, -5.8686273017529835), (-2.747907244501534, -5.988665349966056), (-2.6209967889480175, -6.108703398179129), (-2.4940863333945003, -6.2287414463922035), (-2.367175877840984, -6.348779494605276), (-2.240265422287467, -6.46881754281835), (-2.113354966733951, -6.5888555910314235)}
  19. Sequence(Segment[Element[l4, i], Element[l6, n - i]], i, 1, n) → l7, {5.639994179744307, 5.343533164364803, 5.050468852847458, 4.761428484425344, 4.477191465060505, 4.1987334091512825, 3.9272837548660258, 3.6644003281771678, 3.412063666295544, 3.1727911662071855, 2.9497636735286825, 2.7469408983040413, 2.5691125883620436, 2.4217909299491818, 2.3108179826387993, 2.241598718144816, 2.2180456490543716, 2.2415987181448163, 2.3108179826387985, 2.4217909299491827, 2.5691125883620427, 2.746940898304041, 2.949763673528682, 3.172791166207185, 3.412063666295543, 3.6644003281771664, 3.927283754866023, 4.1987334091512825, 4.477191465060503, 4.761428484425342, 5.050468852847458, 5.3435331643648025, 5.639994179744306}
  20. Segment(A, F) → j
  21. Sequence(A + (i / n * (F - A)), i, 1, n) → l8, {(-6.3037227543085335, -2.8022711240017295), (-6.306045508617068, -2.9769422480034584), (-6.308368262925601, -3.1516133720051878), (-6.310691017234134, -3.3262844960069167), (-6.313013771542669, -3.500955620008646), (-6.315336525851202, -3.6756267440103754), (-6.317659280159736, -3.8502978680121043), (-6.31998203446827, -4.024968992013834), (-6.322304788776803, -4.199640116015563), (-6.324627543085337, -4.374311240017292), (-6.3269502973938705, -4.548982364019022), (-6.329273051702404, -4.72365348802075), (-6.331595806010938, -4.8983246120224795), (-6.333918560319471, -5.072995736024209), (-6.336241314628005, -5.247666860025938), (-6.338564068936539, -5.422337984027667), (-6.340886823245072, -5.597009108029397), (-6.343209577553606, -5.771680232031125), (-6.34553233186214, -5.946351356032855), (-6.347855086170673, -6.121022480034584), (-6.3501778404792075, -6.295693604036313), (-6.352500594787741, -6.470364728038042), (-6.354823349096274, -6.645035852039771), (-6.3571461034048085, -6.8197069760415), (-6.359468857713342, -6.99437810004323), (-6.361791612021875, -7.169049224044959), (-6.364114366330409, -7.343720348046688), (-6.366437120638943, -7.5183914720484175), (-6.368759874947476, -7.693062596050146), (-6.37108262925601, -7.867733720051876), (-6.373405383564544, -8.042404844053605), (-6.375728137873077, -8.217075968055333), (-6.378050892181611, -8.391747092057063)}
  22. Sequence(Segment[Element[l6, i], Element[l8, n - i]], i, 1, n) → l9, {5.6477786891036414, 5.359455269421782, 5.074925480413547, 4.794864717752368, 4.520103745968714, 4.2516701741871294, 3.990840995179148, 3.7392078854365236, 3.498755572370974, 3.2719499817175213, 3.061825271550244, 2.872045000887739, 2.70689160546241, 2.571115132252843, 2.469565764634411, 2.406580206722229, 2.3852154409818342, 2.406580206722231, 2.46956576463441, 2.571115132252843, 2.7068916054624097, 2.8720450008877383, 3.061825271550244, 3.271949981717522, 3.4987555723709742, 3.7392078854365236, 3.99084099517915, 4.251670174187129, 4.520103745968714, 4.794864717752367, 5.074925480413545, 5.359455269421783, 5.6477786891036414}
  23. Segment(A, G) → k
  24. Sequence(A + (i / n * (G - A)), i, 1, n) → l10, {(-6.429984140984296, -2.7458434583177267), (-6.558568281968592, -2.8640869166354537), (-6.687152422952888, -2.98233037495318), (-6.815736563937183, -3.1005738332709067), (-6.94432070492148, -3.2188172915886337), (-7.072904845905775, -3.3370607499063603), (-7.201488986890071, -3.455304208224087), (-7.330073127874368, -3.5735476665418138), (-7.458657268858663, -3.6917911248595403), (-7.587241409842959, -3.8100345831772673), (-7.715825550827255, -3.928278041494994), (-7.844409691811551, -4.04652149981272), (-7.972993832795847, -4.164764958130447), (-8.101577973780142, -4.283008416448174), (-8.230162114764438, -4.401251874765901), (-8.358746255748734, -4.519495333083627), (-8.48733039673303, -4.637738791401354), (-8.615914537717327, -4.7559822497190805), (-8.744498678701621, -4.874225708036807), (-8.873082819685917, -4.9924691663545335), (-9.001666960670214, -5.110712624672261), (-9.13025110165451, -5.2289560829899875), (-9.258835242638806, -5.347199541307714), (-9.3874193836231, -5.465442999625441), (-9.516003524607399, -5.583686457943168), (-9.644587665591693, -5.7019299162608945), (-9.77317180657599, -5.820173374578621), (-9.901755947560286, -5.938416832896348), (-10.03034008854458, -6.056660291214074), (-10.158924229528877, -6.174903749531801), (-10.287508370513173, -6.293147207849527), (-10.41609251149747, -6.411390666167254), (-10.544676652481765, -6.52963412448498)}
  25. Sequence(Segment[Element[l8, i], Element[l10, n - i]], i, 1, n) → l11, {5.646142479941703, 5.356110744086045, 5.069791991548261, 4.7878523899302685, 4.511113090302242, 4.240592323576118, 3.977559092155681, 3.7236004812478387, 3.4807033616568264, 3.251347778274461, 3.0386016511337295, 2.8461919732857357, 2.6785049260475216, 2.540440840576044, 2.437039561766237, 2.3728369354639622, 2.351046480725428, 2.3728369354639622, 2.437039561766237, 2.540440840576044, 2.6785049260475238, 2.846191973285736, 3.03860165113373, 3.2513477782744626, 3.480703361656826, 3.723600481247839, 3.977559092155682, 4.240592323576119, 4.511113090302244, 4.78785238993027, 5.069791991548262, 5.356110744086044, 5.6461424799417035}
  26. Segment(A, H) → l
  27. Sequence(A + (i / n * (H - A)), i, 1, n) → l12, {(-6.476076890042206, -2.6294387041057075), (-6.650753780084412, -2.631277408211415), (-6.825430670126618, -2.6331161123171225), (-7.000107560168824, -2.6349548164228302), (-7.17478445021103, -2.6367935205285375), (-7.349461340253236, -2.638632224634245), (-7.524138230295442, -2.6404709287399526), (-7.698815120337647, -2.64230963284566), (-7.873492010379854, -2.644148336951367), (-8.04816890042206, -2.645987041057075), (-8.222845790464266, -2.6478257451627822), (-8.397522680506471, -2.64966444926849), (-8.572199570548678, -2.6515031533741973), (-8.746876460590883, -2.6533418574799046), (-8.92155335063309, -2.6551805615856123), (-9.096230240675295, -2.6570192656913196), (-9.270907130717502, -2.6588579697970274), (-9.445584020759707, -2.6606966739027347), (-9.620260910801914, -2.662535378008442), (-9.794937800844119, -2.6643740821141497), (-9.969614690886326, -2.666212786219857), (-10.144291580928531, -2.6680514903255648), (-10.318968470970738, -2.669890194431272), (-10.493645361012943, -2.6717288985369794), (-10.668322251055148, -2.673567602642687), (-10.842999141097355, -2.6754063067483944), (-11.017676031139562, -2.6772450108541017), (-11.192352921181767, -2.6790837149598095), (-11.367029811223972, -2.6809224190655168), (-11.541706701266179, -2.6827611231712245), (-11.716383591308386, -2.684599827276932), (-11.89106048135059, -2.686438531382639), (-12.065737371392796, -2.688277235488347)}
  28. Sequence(Segment[Element[l10, i], Element[l12, n - i]], i, 1, n) → l13, {5.636047226919828, 5.335450496935829, 5.038036286468966, 4.744403144721819, 4.45529872543443, 4.171664682523288, 3.8946963516184017, 3.6259215485928986, 3.367302843394704, 3.121365688134104, 2.8913479618217544, 2.681349710676287, 2.4964281409445586, 2.342529532833918, 2.226097418991669, 2.15321823855055, 2.1283706923736725, 2.153218238550549, 2.226097418991668, 2.342529532833917, 2.496428140944559, 2.681349710676287, 2.8913479618217526, 3.121365688134104, 3.3673028433947056, 3.6259215485928995, 3.8946963516184003, 4.1716646825232875, 4.455298725434429, 4.744403144721818, 5.038036286468964, 5.33545049693583, 5.636047226919829}
  29. Segment(A, I) → m
  30. Sequence(A + (i / n * (I - A)), i, 1, n) → l14, {(-6.42605417550702, -2.50522053572486), (-6.55070835101404, -2.38284107144972), (-6.67536252652106, -2.26046160717458), (-6.80001670202808, -2.1380821428994397), (-6.9246708775351005, -2.015702678624299), (-7.0493250530421205, -1.893323214349159), (-7.17397922854914, -1.7709437500740188), (-7.29863340405616, -1.6485642857988787), (-7.42328757956318, -1.5261848215237386), (-7.5479417550702, -1.4038053572485985), (-7.67259593057722, -1.281425892973458), (-7.79725010608424, -1.159046428698318), (-7.92190428159126, -1.0366669644231776), (-8.04655845709828, -0.9142875001480377), (-8.1712126326053, -0.7919080358728974), (-8.29586680811232, -0.6695285715977572), (-8.42052098361934, -0.5471491073226171), (-8.54517515912636, -0.424769643047477), (-8.66982933463338, -0.30239017877233687), (-8.7944835101404, -0.18001071449719674), (-8.91913768564742, -0.057631250222056174), (-9.04379186115444, 0.06474821405308395), (-9.16844603666146, 0.18712767832822408), (-9.293100212168481, 0.3095071426033642), (-9.417754387675501, 0.4318866068785048), (-9.542408563182521, 0.5542660711536449), (-9.667062738689541, 0.676645535428785), (-9.79171691419656, 0.7990249997039247), (-9.916371089703581, 0.9214044639790648), (-10.041025265210601, 1.0437839282542054), (-10.165679440717621, 1.1661633925293455), (-10.29033361622464, 1.2885428568044857), (-10.414987791731662, 1.4109223210796262)}
  31. Sequence(Segment[Element[l12, i], Element[l14, n - i]], i, 1, n) → l15, {5.642653303658193, 5.348974882504858, 5.0588325338343765, 4.772871173667326, 4.491889379626131, 4.216882673314689, 3.949099528441313, 3.690112872785772, 3.4419089566101615, 3.206992335603151, 2.9884980741120737, 2.790286716568016, 2.6169710555660295, 2.4737892743754384, 2.3662179877791605, 2.2992607683110853, 2.2765041003761497, 2.2992607683110853, 2.3662179877791614, 2.4737892743754384, 2.61697105556603, 2.790286716568015, 2.9884980741120746, 3.20699233560315, 3.4419089566101593, 3.6901128727857717, 3.949099528441313, 4.216882673314689, 4.491889379626128, 4.772871173667325, 5.0588325338343765, 5.348974882504856, 5.642653303658191}
  32. Sequence(Segment[Element[l14, i], Element[l1, n - i]], i, 1, n) → l16, {5.643654069263912, 5.351022112439773, 5.061977659288176, 4.77717194367742, 4.4974103122572275, 4.223695187291514, 3.957281432311104, 3.699746669015573, 3.4530780871973232, 3.2197740477688335, 3.002951105513035, 2.8064318175453864, 2.634763278166205, 2.4930842942885096, 2.386741470267743, 2.320597805389056, 2.298126907694626, 2.320597805389056, 2.3867414702677436, 2.49308429428851, 2.634763278166205, 2.8064318175453873, 3.002951105513035, 3.2197740477688344, 3.4530780871973237, 3.6997466690155743, 3.9572814323111043, 4.223695187291512, 4.4974103122572275, 4.77717194367742, 5.0619776592881776, 5.351022112439773, 5.6436540692639126}

AI Drawing Prompt Reference

Create a decorative geometric pattern centered on a single circle. Start by selecting a base segment with two given endpoints, then define a circle using those two points as inputs, and place multiple points on the circle. Use repeated interpolation between the base point and several other circle-based points to generate several ordered sets of points along different directions. Connect corresponding point pairs to form many short curve-like border segments, and extend the pattern by sampling distances in a mirrored order between paired segments so the motif repeats symmetrically across the design. The result should look like festive layered bands wrapping around the central circle, leaving space to add holiday text and an image.
The prompt above was compiled from the original drawing commands and is for reference only.