Buffon's Needle Simulation: Empirical Method for Estimating Pi
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Introduction to Probability
① This example simulates Buffon's needle experiment, demonstrating how the probability of a randomly dropped needle crossing parallel lines relates to the value of π. ② Core mathematical concepts include geometric probability, random simulation, frequency versus probability, and numerical estimation of π. ③ Construction highlights: using GeoGebra commands such as RandomUniform to generate random positions and angles, Segment to create needles, Point for endpoints, OrthogonalLine for parallel lines, Sum and mean to compute crossing counts and frequencies, Distance to measure needle length and line spacing, Intersect to detect crossings, Vector for direction, and Name for labeling; multiple objects (A501 to A496) accumulate data over many trials. ④ Suitable for high school and above, in probability, statistics, or mathematical experimentation, helping students intuitively grasp Monte Carlo methods and approximate computation of π.
GeoGebra Commands & Parameters
RandomUniform((-π), π) → B501RandomUniform((-π), π) → B500RandomUniform((-π), π) → B499RandomUniform((-π), π) → B498RandomUniform((-π), π) → B497RandomUniform((-π), π) → B496RandomUniform((-π), π) → B495RandomUniform((-π), π) → B494RandomUniform((-π), π) → B493RandomUniform((-π), π) → B492RandomUniform((-π), π) → B491RandomUniform((-π), π) → B490RandomUniform((-π), π) → B489RandomUniform((-π), π) → B488RandomUniform((-π), π) → B487RandomUniform((-π), π) → B486RandomUniform((-π), π) → B485RandomUniform((-π), π) → B484RandomUniform((-π), π) → B483RandomUniform((-π), π) → B482RandomUniform((-π), π) → B481RandomUniform((-π), π) → B480RandomUniform((-π), π) → B479RandomUniform((-π), π) → B478RandomUniform((-π), π) → B477RandomUniform((-π), π) → B476RandomUniform((-π), π) → B475RandomUniform((-π), π) → B474RandomUniform((-π), π) → B473RandomUniform((-π), π) → B472RandomUniform((-π), π) → B471RandomUniform((-π), π) → B470RandomUniform((-π), π) → B469RandomUniform((-π), π) → B468RandomUniform((-π), π) → B467RandomUniform((-π), π) → B466RandomUniform((-π), π) → B465RandomUniform((-π), π) → B464RandomUniform((-π), π) → B463RandomUniform((-π), π) → B462RandomUniform((-π), π) → B461RandomUniform((-π), π) → B460RandomUniform((-π), π) → B459RandomUniform((-π), π) → B458RandomUniform((-π), π) → B457RandomUniform((-π), π) → B456RandomUniform((-π), π) → B455RandomUniform((-π), π) → B454RandomUniform((-π), π) → B453RandomUniform((-π), π) → B452RandomUniform((-π), π) → B451RandomUniform((-π), π) → B450RandomUniform((-π), π) → B449RandomUniform((-π), π) → B448RandomUniform((-π), π) → B447RandomUniform((-π), π) → B446RandomUniform((-π), π) → B445RandomUniform((-π), π) → B444RandomUniform((-π), π) → B443RandomUniform((-π), π) → B442RandomUniform((-π), π) → B441RandomUniform((-π), π) → B440RandomUniform((-π), π) → B439RandomUniform((-π), π) → B438RandomUniform((-π), π) → B437RandomUniform((-π), π) → B436RandomUniform((-π), π) → B435RandomUniform((-π), π) → B434RandomUniform((-π), π) → B433RandomUniform((-π), π) → B432RandomUniform((-π), π) → B431RandomUniform((-π), π) → B430RandomUniform((-π), π) → B429RandomUniform((-π), π) → B428RandomUniform((-π), π) → B427RandomUniform((-π), π) → B426RandomUniform((-π), π) → B425RandomUniform((-π), π) → B424RandomUniform((-π), π) → B423RandomUniform((-π), π) → B422RandomUniform((-π), π) → B421RandomUniform((-π), π) → B420RandomUniform((-π), π) → B419RandomUniform((-π), π) → B418RandomUniform((-π), π) → B417RandomUniform((-π), π) → B416RandomUniform((-π), π) → B415RandomUniform((-π), π) → B414RandomUniform((-π), π) → B413RandomUniform((-π), π) → B412RandomUniform((-π), π) → B411RandomUniform((-π), π) → B410RandomUniform((-π), π) → B409RandomUniform((-π), π) → B408RandomUniform((-π), π) → B407RandomUniform((-π), π) → B406RandomUniform((-π), π) → B405RandomUniform((-π), π) → B404RandomUniform((-π), π) → B403RandomUniform((-π), π) → B402RandomUniform((-π), π) → B401RandomUniform((-π), π) → B400RandomUniform((-π), π) → B399RandomUniform((-π), π) → B398RandomUniform((-π), π) → B397RandomUniform((-π), π) → B396RandomUniform((-π), π) → B395RandomUniform((-π), π) → B394RandomUniform((-π), π) → B393RandomUniform((-π), π) → B392RandomUniform((-π), π) → B391RandomUniform((-π), π) → B390RandomUniform((-π), π) → B389RandomUniform((-π), π) → B388RandomUniform((-π), π) → B387RandomUniform((-π), π) → B386RandomUniform((-π), π) → B385RandomUniform((-π), π) → B384RandomUniform((-π), π) → B383RandomUniform((-π), π) → B382RandomUniform((-π), π) → B381RandomUniform((-π), π) → B380RandomUniform((-π), π) → B379RandomUniform((-π), π) → B378RandomUniform((-π), π) → B377RandomUniform((-π), π) → B376RandomUniform((-π), π) → B375RandomUniform((-π), π) → B374RandomUniform((-π), π) → B373RandomUniform((-π), π) → B372RandomUniform((-π), π) → B371RandomUniform((-π), π) → B370RandomUniform((-π), π) → B369RandomUniform((-π), π) → B368RandomUniform((-π), π) → B367RandomUniform((-π), π) → B366RandomUniform((-π), π) → B365RandomUniform((-π), π) → B364RandomUniform((-π), π) → B363RandomUniform((-π), π) → B362RandomUniform((-π), π) → B361RandomUniform((-π), π) → B360RandomUniform((-π), π) → B359RandomUniform((-π), π) → B358RandomUniform((-π), π) → B357RandomUniform((-π), π) → B356RandomUniform((-π), π) → B355RandomUniform((-π), π) → B354RandomUniform((-π), π) → B353RandomUniform((-π), π) → B352RandomUniform((-π), π) → B351RandomUniform((-π), π) → B350RandomUniform((-π), π) → B349RandomUniform((-π), π) → B348RandomUniform((-π), π) → B347RandomUniform((-π), π) → B346RandomUniform((-π), π) → B345RandomUniform((-π), π) → B344RandomUniform((-π), π) → B343RandomUniform((-π), π) → B342RandomUniform((-π), π) → B341RandomUniform((-π), π) → B340RandomUniform((-π), π) → B339RandomUniform((-π), π) → B338RandomUniform((-π), π) → B337RandomUniform((-π), π) → B336RandomUniform((-π), π) → B335RandomUniform((-π), π) → B334RandomUniform((-π), π) → B333RandomUniform((-π), π) → B332RandomUniform((-π), π) → B331RandomUniform((-π), π) → B330RandomUniform((-π), π) → B329RandomUniform((-π), π) → B328RandomUniform((-π), π) → B327RandomUniform((-π), π) → B326RandomUniform((-π), π) → B325RandomUniform((-π), π) → B324RandomUniform((-π), π) → B323RandomUniform((-π), π) → B322RandomUniform((-π), π) → B321RandomUniform((-π), π) → B320RandomUniform((-π), π) → B319RandomUniform((-π), π) → B318RandomUniform((-π), π) → B317RandomUniform((-π), π) → B316RandomUniform((-π), π) → B315RandomUniform((-π), π) → B314RandomUniform((-π), π) → B313RandomUniform((-π), π) → B312RandomUniform((-π), π) → B311RandomUniform((-π), π) → B310RandomUniform((-π), π) → B309RandomUniform((-π), π) → B308RandomUniform((-π), π) → B307RandomUniform((-π), π) → B306RandomUniform((-π), π) → B305RandomUniform((-π), π) → B304RandomUniform((-π), π) → B303RandomUniform((-π), π) → B302RandomUniform((-π), π) → B301RandomUniform((-π), π) → B300RandomUniform((-π), π) → B299RandomUniform((-π), π) → B298RandomUniform((-π), π) → B297RandomUniform((-π), π) → B296RandomUniform((-π), π) → B295RandomUniform((-π), π) → B294RandomUniform((-π), π) → B293RandomUniform((-π), π) → B292RandomUniform((-π), π) → B291RandomUniform((-π), π) → B290RandomUniform((-π), π) → B289RandomUniform((-π), π) → B288RandomUniform((-π), π) → B287RandomUniform((-π), π) → B286RandomUniform((-π), π) → B285RandomUniform((-π), π) → B284RandomUniform((-π), π) → B283RandomUniform((-π), π) → B282RandomUniform((-π), π) → B281RandomUniform((-π), π) → B280RandomUniform((-π), π) → B279RandomUniform((-π), π) → B278RandomUniform((-π), π) → B277RandomUniform((-π), π) → B276RandomUniform((-π), π) → B275RandomUniform((-π), π) → B274RandomUniform((-π), π) → B273RandomUniform((-π), π) → B272RandomUniform((-π), π) → B271RandomUniform((-π), π) → B270RandomUniform((-π), π) → B269RandomUniform((-π), π) → B268RandomUniform((-π), π) → B267RandomUniform((-π), π) → B266RandomUniform((-π), π) → B265RandomUniform((-π), π) → B264RandomUniform((-π), π) → B263RandomUniform((-π), π) → B262RandomUniform((-π), π) → B261RandomUniform((-π), π) → B260RandomUniform((-π), π) → B259RandomUniform((-π), π) → B258RandomUniform((-π), π) → B257RandomUniform((-π), π) → B256RandomUniform((-π), π) → B255RandomUniform((-π), π) → B254RandomUniform((-π), π) → B253RandomUniform((-π), π) → B252RandomUniform((-π), π) → B251RandomUniform((-π), π) → B250RandomUniform((-π), π) → B249RandomUniform((-π), π) → B248RandomUniform((-π), π) → B247RandomUniform((-π), π) → B246RandomUniform((-π), π) → B245RandomUniform((-π), π) → B244RandomUniform((-π), π) → B243RandomUniform((-π), π) → B242RandomUniform((-π), π) → B241RandomUniform((-π), π) → B240RandomUniform((-π), π) → B239RandomUniform((-π), π) → B238RandomUniform((-π), π) → B237RandomUniform((-π), π) → B236RandomUniform((-π), π) → B235RandomUniform((-π), π) → B234RandomUniform((-π), π) → B233RandomUniform((-π), π) → B232RandomUniform((-π), π) → B231RandomUniform((-π), π) → B230RandomUniform((-π), π) → B229RandomUniform((-π), π) → B228RandomUniform((-π), π) → B227RandomUniform((-π), π) → B226RandomUniform((-π), π) → B225RandomUniform((-π), π) → B224RandomUniform((-π), π) → B223RandomUniform((-π), π) → B222RandomUniform((-π), π) → B221RandomUniform((-π), π) → B220RandomUniform((-π), π) → B219RandomUniform((-π), π) → B218RandomUniform((-π), π) → B217RandomUniform((-π), π) → B216RandomUniform((-π), π) → B215RandomUniform((-π), π) → B214RandomUniform((-π), π) → B213RandomUniform((-π), π) → B212RandomUniform((-π), π) → B211RandomUniform((-π), π) → B210RandomUniform((-π), π) → B209RandomUniform((-π), π) → B208RandomUniform((-π), π) → B207RandomUniform((-π), π) → B206RandomUniform((-π), π) → B205RandomUniform((-π), π) → B204RandomUniform((-π), π) → B203RandomUniform((-π), π) → B202RandomUniform((-π), π) → B2Segment(C2, D2) → E2RandomUniform((-π), π) → B3Segment(C3, D3) → E3RandomUniform((-π), π) → B4Segment(C4, D4) → E4RandomUniform((-π), π) → B5Segment(C5, D5) → E5RandomUniform((-π), π) → B6Segment(C6, D6) → E6RandomUniform((-π), π) → B7Segment(C7, D7) → E7RandomUniform((-π), π) → B8Segment(C8, D8) → E8RandomUniform((-π), π) → B9Segment(C9, D9) → E9RandomUniform((-π), π) → B10Segment(C10, D10) → E10RandomUniform((-π), π) → B11Segment(C11, D11) → E11RandomUniform((-π), π) → B12Segment(C12, D12) → E12RandomUniform((-π), π) → B13Segment(C13, D13) → E13RandomUniform((-π), π) → B14Segment(C14, D14) → E14RandomUniform((-π), π) → B15Segment(C15, D15) → E15RandomUniform((-π), π) → B16Segment(C16, D16) → E16RandomUniform((-π), π) → B17Segment(C17, D17) → E17RandomUniform((-π), π) → B18Segment(C18, D18) → E18RandomUniform((-π), π) → B19Segment(C19, D19) → E19RandomUniform((-π), π) → B20Segment(C20, D20) → E20RandomUniform((-π), π) → B21Segment(C21, D21) → E21RandomUniform((-π), π) → B22Segment(C22, D22) → E22RandomUniform((-π), π) → B23Segment(C23, D23) → E23RandomUniform((-π), π) → B24Segment(C24, D24) → E24RandomUniform((-π), π) → B25Segment(C25, D25) → E25RandomUniform((-π), π) → B26Segment(C26, D26) → E26Point(yAxis) → APoint(xAxis) → BPoint(xAxis) → CPoint(xAxis) → DPoint(xAxis) → EPoint(xAxis) → FOrthogonalLine(A, xAxis) → fOrthogonalLine(B, xAxis) → gOrthogonalLine(C, xAxis) → hOrthogonalLine(D, xAxis) → iOrthogonalLine(E, xAxis) → jOrthogonalLine(F, xAxis) → kRandomUniform((-π), π) → B27Segment(C27, D27) → E27RandomUniform((-π), π) → B28Segment(C28, D28) → E28RandomUniform((-π), π) → B29Segment(C29, D29) → E29RandomUniform((-π), π) → B30Segment(C30, D30) → E30RandomUniform((-π), π) → B31Segment(C31, D31) → E31RandomUniform((-π), π) → B32Segment(C32, D32) → E32RandomUniform((-π), π) → B33Segment(C33, D33) → E33RandomUniform((-π), π) → B34Segment(C34, D34) → E34RandomUniform((-π), π) → B35Segment(C35, D35) → E35RandomUniform((-π), π) → B36Segment(C36, D36) → E36RandomUniform((-π), π) → B37Segment(C37, D37) → E37RandomUniform((-π), π) → B38Segment(C38, D38) → E38RandomUniform((-π), π) → B39Segment(C39, D39) → E39RandomUniform((-π), π) → B40Segment(C40, D40) → E40RandomUniform((-π), π) → B41Segment(C41, D41) → E41RandomUniform((-π), π) → B42Segment(C42, D42) → E42RandomUniform((-π), π) → B43Segment(C43, D43) → E43RandomUniform((-π), π) → B44Segment(C44, D44) → E44RandomUniform((-π), π) → B45Segment(C45, D45) → E45RandomUniform((-π), π) → B46Segment(C46, D46) → E46RandomUniform((-π), π) → B47Segment(C47, D47) → E47RandomUniform((-π), π) → B48Segment(C48, D48) → E48RandomUniform((-π), π) → B49Segment(C49, D49) → E49RandomUniform((-π), π) → B50Segment(C50, D50) → E50RandomUniform((-π), π) → B51Segment(C51, D51) → E51RandomUniform((-π), π) → B52Segment(C52, D52) → E52RandomUniform((-π), π) → B53Segment(C53, D53) → E53RandomUniform((-π), π) → B54Segment(C54, D54) → E54RandomUniform((-π), π) → B55Segment(C55, D55) → E55RandomUniform((-π), π) → B56Segment(C56, D56) → E56RandomUniform((-π), π) → B57Segment(C57, D57) → E57RandomUniform((-π), π) → B58Segment(C58, D58) → E58RandomUniform((-π), π) → B59Segment(C59, D59) → E59RandomUniform((-π), π) → B60Segment(C60, D60) → E60RandomUniform((-π), π) → B61Segment(C61, D61) → E61RandomUniform((-π), π) → B62Segment(C62, D62) → E62RandomUniform((-π), π) → B63Segment(C63, D63) → E63RandomUniform((-π), π) → B64Segment(C64, D64) → E64RandomUniform((-π), π) → B65Segment(C65, D65) → E65RandomUniform((-π), π) → B66Segment(C66, D66) → E66RandomUniform((-π), π) → B67Segment(C67, D67) → E67RandomUniform((-π), π) → B68Segment(C68, D68) → E68RandomUniform((-π), π) → B69Segment(C69, D69) → E69RandomUniform((-π), π) → B70Segment(C70, D70) → E70RandomUniform((-π), π) → B71Segment(C71, D71) → E71RandomUniform((-π), π) → B72Segment(C72, D72) → E72RandomUniform((-π), π) → B73Segment(C73, D73) → E73RandomUniform((-π), π) → B74Segment(C74, D74) → E74RandomUniform((-π), π) → B75Segment(C75, D75) → E75RandomUniform((-π), π) → B76Segment(C76, D76) → E76RandomUniform((-π), π) → B77Segment(C77, D77) → E77RandomUniform((-π), π) → B78Segment(C78, D78) → E78RandomUniform((-π), π) → B79Segment(C79, D79) → E79RandomUniform((-π), π) → B80Segment(C80, D80) → E80RandomUniform((-π), π) → B81Segment(C81, D81) → E81RandomUniform((-π), π) → B82Segment(C82, D82) → E82RandomUniform((-π), π) → B83Segment(C83, D83) → E83RandomUniform((-π), π) → B84Segment(C84, D84) → E84RandomUniform((-π), π) → B85Segment(C85, D85) → E85RandomUniform((-π), π) → B86Segment(C86, D86) → E86RandomUniform((-π), π) → B87Segment(C87, D87) → E87RandomUniform((-π), π) → B88Segment(C88, D88) → E88RandomUniform((-π), π) → B89Segment(C89, D89) → E89RandomUniform((-π), π) → B90Segment(C90, D90) → E90RandomUniform((-π), π) → B91Segment(C91, D91) → E91RandomUniform((-π), π) → B92Segment(C92, D92) → E92RandomUniform((-π), π) → B93Segment(C93, D93) → E93RandomUniform((-π), π) → B94Segment(C94, D94) → E94RandomUniform((-π), π) → B95Segment(C95, D95) → E95RandomUniform((-π), π) → B96Segment(C96, D96) → E96RandomUniform((-π), π) → B97Segment(C97, D97) → E97RandomUniform((-π), π) → B98Segment(C98, D98) → E98RandomUniform((-π), π) → B99Segment(C99, D99) → E99RandomUniform((-π), π) → B100Segment(C100, D100) → E100RandomUniform((-π), π) → B101Segment(C101, D101) → E101RandomUniform((-π), π) → B102Segment(C102, D102) → E102RandomUniform((-π), π) → B103Segment(C103, D103) → E103RandomUniform((-π), π) → B104Segment(C104, D104) → E104RandomUniform((-π), π) → B105Segment(C105, D105) → E105RandomUniform((-π), π) → B106Segment(C106, D106) → E106RandomUniform((-π), π) → B107Segment(C107, D107) → E107RandomUniform((-π), π) → B108Segment(C108, D108) → E108RandomUniform((-π), π) → B109Segment(C109, D109) → E109RandomUniform((-π), π) → B110Segment(C110, D110) → E110RandomUniform((-π), π) → B111Segment(C111, D111) → E111RandomUniform((-π), π) → B112Segment(C112, D112) → E112RandomUniform((-π), π) → B113Segment(C113, D113) → E113RandomUniform((-π), π) → B114Segment(C114, D114) → E114RandomUniform((-π), π) → B115Segment(C115, D115) → E115RandomUniform((-π), π) → B116Segment(C116, D116) → E116RandomUniform((-π), π) → B117Segment(C117, D117) → E117RandomUniform((-π), π) → B118Segment(C118, D118) → E118RandomUniform((-π), π) → B119Segment(C119, D119) → E119RandomUniform((-π), π) → B120Segment(C120, D120) → E120RandomUniform((-π), π) → B121Segment(C121, D121) → E121RandomUniform((-π), π) → B122Segment(C122, D122) → E122RandomUniform((-π), π) → B123Segment(C123, D123) → E123RandomUniform((-π), π) → B124Segment(C124, D124) → E124RandomUniform((-π), π) → B125Segment(C125, D125) → E125RandomUniform((-π), π) → B126Segment(C126, D126) → E126RandomUniform((-π), π) → B127Segment(C127, D127) → E127RandomUniform((-π), π) → B128Segment(C128, D128) → E128RandomUniform((-π), π) → B129Segment(C129, D129) → E129RandomUniform((-π), π) → B130Segment(C130, D130) → E130RandomUniform((-π), π) → B131Segment(C131, D131) → E131RandomUniform((-π), π) → B132Segment(C132, D132) → E132RandomUniform((-π), π) → B133Segment(C133, D133) → E133RandomUniform((-π), π) → B134Segment(C134, D134) → E134RandomUniform((-π), π) → B135Segment(C135, D135) → E135RandomUniform((-π), π) → B136Segment(C136, D136) → E136RandomUniform((-π), π) → B137Segment(C137, D137) → E137RandomUniform((-π), π) → B138Segment(C138, D138) → E138RandomUniform((-π), π) → B139Segment(C139, D139) → E139RandomUniform((-π), π) → B140Segment(C140, D140) → E140RandomUniform((-π), π) → B141Segment(C141, D141) → E141RandomUniform((-π), π) → B142Segment(C142, D142) → E142RandomUniform((-π), π) → B143Segment(C143, D143) → E143RandomUniform((-π), π) → B144Segment(C144, D144) → E144RandomUniform((-π), π) → B145Segment(C145, D145) → E145RandomUniform((-π), π) → B146Segment(C146, D146) → E146RandomUniform((-π), π) → B147Segment(C147, D147) → E147RandomUniform((-π), π) → B148Segment(C148, D148) → E148RandomUniform((-π), π) → B149Segment(C149, D149) → E149RandomUniform((-π), π) → B150Segment(C150, D150) → E150RandomUniform((-π), π) → B151Segment(C151, D151) → E151RandomUniform((-π), π) → B152Segment(C152, D152) → E152RandomUniform((-π), π) → B153Segment(C153, D153) → E153RandomUniform((-π), π) → B154Segment(C154, D154) → E154RandomUniform((-π), π) → B155Segment(C155, D155) → E155RandomUniform((-π), π) → B156Segment(C156, D156) → E156RandomUniform((-π), π) → B157Segment(C157, D157) → E157RandomUniform((-π), π) → B158Segment(C158, D158) → E158RandomUniform((-π), π) → B159Segment(C159, D159) → E159RandomUniform((-π), π) → B160Segment(C160, D160) → E160RandomUniform((-π), π) → B161Segment(C161, D161) → E161RandomUniform((-π), π) → B162Segment(C162, D162) → E162RandomUniform((-π), π) → B163Segment(C163, D163) → E163RandomUniform((-π), π) → B164Segment(C164, D164) → E164RandomUniform((-π), π) → B165Segment(C165, D165) → E165RandomUniform((-π), π) → B166Segment(C166, D166) → E166RandomUniform((-π), π) → B167Segment(C167, D167) → E167RandomUniform((-π), π) → B168Segment(C168, D168) → E168RandomUniform((-π), π) → B169Segment(C169, D169) → E169RandomUniform((-π), π) → B170Segment(C170, D170) → E170RandomUniform((-π), π) → B171Segment(C171, D171) → E171RandomUniform((-π), π) → B172Segment(C172, D172) → E172RandomUniform((-π), π) → B173Segment(C173, D173) → E173RandomUniform((-π), π) → B174Segment(C174, D174) → E174RandomUniform((-π), π) → B175Segment(C175, D175) → E175RandomUniform((-π), π) → B176Segment(C176, D176) → E176RandomUniform((-π), π) → B177Segment(C177, D177) → E177RandomUniform((-π), π) → B178Segment(C178, D178) → E178RandomUniform((-π), π) → B179Segment(C179, D179) → E179RandomUniform((-π), π) → B180Segment(C180, D180) → E180RandomUniform((-π), π) → B181Segment(C181, D181) → E181RandomUniform((-π), π) → B182Segment(C182, D182) → E182RandomUniform((-π), π) → B183Segment(C183, D183) → E183RandomUniform((-π), π) → B184Segment(C184, D184) → E184RandomUniform((-π), π) → B185Segment(C185, D185) → E185RandomUniform((-π), π) → B186Segment(C186, D186) → E186RandomUniform((-π), π) → B187Segment(C187, D187) → E187RandomUniform((-π), π) → B188Segment(C188, D188) → E188RandomUniform((-π), π) → B189Segment(C189, D189) → E189RandomUniform((-π), π) → B190Segment(C190, D190) → E190RandomUniform((-π), π) → B191Segment(C191, D191) → E191RandomUniform((-π), π) → B192Segment(C192, D192) → E192RandomUniform((-π), π) → B193Segment(C193, D193) → E193RandomUniform((-π), π) → B194Segment(C194, D194) → E194RandomUniform((-π), π) → B195Segment(C195, D195) → E195RandomUniform((-π), π) → B196Segment(C196, D196) → E196RandomUniform((-π), π) → B197Segment(C197, D197) → E197RandomUniform((-π), π) → B198Segment(C198, D198) → E198RandomUniform((-π), π) → B199Segment(C199, D199) → E199RandomUniform((-π), π) → B200Segment(C200, D200) → E200RandomUniform((-π), π) → B201Segment(C201, D201) → E201Sum(I2:I501) → aSegment(C202, D202) → E202Segment(C203, D203) → E203Segment(C204, D204) → E204Segment(C205, D205) → E205Segment(C206, D206) → E206Segment(C207, D207) → E207Segment(C208, D208) → E208Segment(C209, D209) → E209Segment(C210, D210) → E210Segment(C211, D211) → E211Segment(C212, D212) → E212Segment(C213, D213) → E213Segment(C214, D214) → E214Segment(C215, D215) → E215Segment(C216, D216) → E216Segment(C217, D217) → E217Segment(C218, D218) → E218Segment(C219, D219) → E219Segment(C220, D220) → E220Segment(C221, D221) → E221Segment(C222, D222) → E222Segment(C223, D223) → E223Segment(C224, D224) → E224Segment(C225, D225) → E225Segment(C226, D226) → E226Segment(C227, D227) → E227Segment(C228, D228) → E228Segment(C229, D229) → E229Segment(C230, D230) → E230Segment(C231, D231) → E231Segment(C232, D232) → E232Segment(C233, D233) → E233Segment(C234, D234) → E234Segment(C235, D235) → E235Segment(C236, D236) → E236Segment(C237, D237) → E237Segment(C238, D238) → E238Segment(C239, D239) → E239Segment(C240, D240) → E240Segment(C241, D241) → E241Segment(C242, D242) → E242Segment(C243, D243) → E243Segment(C244, D244) → E244Segment(C245, D245) → E245Segment(C246, D246) → E246Segment(C247, D247) → E247Segment(C248, D248) → E248Segment(C249, D249) → E249Segment(C250, D250) → E250Segment(C251, D251) → E251Segment(C252, D252) → E252Segment(C253, D253) → E253Segment(C254, D254) → E254Segment(C255, D255) → E255Segment(C256, D256) → E256Segment(C257, D257) → E257Segment(C258, D258) → E258Segment(C259, D259) → E259Segment(C260, D260) → E260Segment(C261, D261) → E261Segment(C262, D262) → E262Segment(C263, D263) → E263Segment(C264, D264) → E264Segment(C265, D265) → E265Segment(C266, D266) → E266Segment(C267, D267) → E267Segment(C268, D268) → E268Segment(C269, D269) → E269Segment(C270, D270) → E270Segment(C271, D271) → E271Segment(C272, D272) → E272Segment(C273, D273) → E273Segment(C274, D274) → E274Segment(C275, D275) → E275Segment(C276, D276) → E276Segment(C277, D277) → E277Segment(C278, D278) → E278Segment(C279, D279) → E279Segment(C280, D280) → E280Segment(C281, D281) → E281Segment(C282, D282) → E282Segment(C283, D283) → E283Segment(C284, D284) → E284Segment(C285, D285) → E285Segment(C286, D286) → E286Segment(C287, D287) → E287Segment(C288, D288) → E288Segment(C289, D289) → E289Segment(C290, D290) → E290Segment(C291, D291) → E291Segment(C292, D292) → E292Segment(C293, D293) → E293Segment(C294, D294) → E294Segment(C295, D295) → E295Segment(C296, D296) → E296Segment(C297, D297) → E297Segment(C298, D298) → E298Segment(C299, D299) → E299Segment(C300, D300) → E300Segment(C301, D301) → E301Segment(C302, D302) → E302Segment(C303, D303) → E303Segment(C304, D304) → E304Segment(C305, D305) → E305Segment(C306, D306) → E306Segment(C307, D307) → E307Segment(C308, D308) → E308Segment(C309, D309) → E309Segment(C310, D310) → E310Segment(C311, D311) → E311Segment(C312, D312) → E312Segment(C313, D313) → E313Segment(C314, D314) → E314Segment(C315, D315) → E315Segment(C316, D316) → E316Segment(C317, D317) → E317Segment(C318, D318) → E318Segment(C319, D319) → E319Segment(C320, D320) → E320Segment(C321, D321) → E321Segment(C322, D322) → E322Segment(C323, D323) → E323Segment(C324, D324) → E324Segment(C325, D325) → E325Segment(C326, D326) → E326Segment(C327, D327) → E327Segment(C328, D328) → E328Segment(C329, D329) → E329Segment(C330, D330) → E330Segment(C331, D331) → E331Segment(C332, D332) → E332Segment(C333, D333) → E333Segment(C334, D334) → E334Segment(C335, D335) → E335Segment(C336, D336) → E336Segment(C337, D337) → E337Segment(C338, D338) → E338Segment(C339, D339) → E339Segment(C340, D340) → E340Segment(C341, D341) → E341Segment(C342, D342) → E342Segment(C343, D343) → E343Segment(C344, D344) → E344Segment(C345, D345) → E345Segment(C346, D346) → E346Segment(C347, D347) → E347Segment(C348, D348) → E348Segment(C349, D349) → E349Segment(C350, D350) → E350Segment(C351, D351) → E351Segment(C352, D352) → E352Segment(C353, D353) → E353Segment(C354, D354) → E354Segment(C355, D355) → E355Segment(C356, D356) → E356Segment(C357, D357) → E357Segment(C358, D358) → E358Segment(C359, D359) → E359Segment(C360, D360) → E360Segment(C361, D361) → E361Segment(C362, D362) → E362Segment(C363, D363) → E363Segment(C364, D364) → E364Segment(C365, D365) → E365Segment(C366, D366) → E366Segment(C367, D367) → E367Segment(C368, D368) → E368Segment(C369, D369) → E369Segment(C370, D370) → E370Segment(C371, D371) → E371Segment(C372, D372) → E372Segment(C373, D373) → E373Segment(C374, D374) → E374Segment(C375, D375) → E375Segment(C376, D376) → E376Segment(C377, D377) → E377Segment(C378, D378) → E378Segment(C379, D379) → E379Segment(C380, D380) → E380Segment(C381, D381) → E381Segment(C382, D382) → E382Segment(C383, D383) → E383Segment(C384, D384) → E384Segment(C385, D385) → E385Segment(C386, D386) → E386Segment(C387, D387) → E387Segment(C388, D388) → E388Segment(C389, D389) → E389Segment(C390, D390) → E390Segment(C391, D391) → E391Segment(C392, D392) → E392Segment(C393, D393) → E393Segment(C394, D394) → E394Segment(C395, D395) → E395Segment(C396, D396) → E396Segment(C397, D397) → E397Segment(C398, D398) → E398Segment(C399, D399) → E399Segment(C400, D400) → E400Segment(C401, D401) → E401Segment(C402, D402) → E402Segment(C403, D403) → E403Segment(C404, D404) → E404Segment(C405, D405) → E405Segment(C406, D406) → E406Segment(C407, D407) → E407Segment(C408, D408) → E408Segment(C409, D409) → E409Segment(C410, D410) → E410Segment(C411, D411) → E411Segment(C412, D412) → E412Segment(C413, D413) → E413Segment(C414, D414) → E414Segment(C415, D415) → E415Segment(C416, D416) → E416Segment(C417, D417) → E417Segment(C418, D418) → E418Segment(C419, D419) → E419Segment(C420, D420) → E420Segment(C421, D421) → E421Segment(C422, D422) → E422Segment(C423, D423) → E423Segment(C424, D424) → E424Segment(C425, D425) → E425Segment(C426, D426) → E426Segment(C427, D427) → E427Segment(C428, D428) → E428Segment(C429, D429) → E429Segment(C430, D430) → E430Segment(C431, D431) → E431Segment(C432, D432) → E432Segment(C433, D433) → E433Segment(C434, D434) → E434Segment(C435, D435) → E435Segment(C436, D436) → E436Segment(C437, D437) → E437Segment(C438, D438) → E438Segment(C439, D439) → E439Segment(C440, D440) → E440Segment(C441, D441) → E441Segment(C442, D442) → E442Segment(C443, D443) → E443Segment(C444, D444) → E444Segment(C445, D445) → E445Segment(C446, D446) → E446Segment(C447, D447) → E447Segment(C448, D448) → E448Segment(C449, D449) → E449Segment(C450, D450) → E450Segment(C451, D451) → E451Segment(C452, D452) → E452Segment(C453, D453) → E453Segment(C454, D454) → E454Segment(C455, D455) → E455Segment(C456, D456) → E456Segment(C457, D457) → E457Segment(C458, D458) → E458Segment(C459, D459) → E459Segment(C460, D460) → E460Segment(C461, D461) → E461Segment(C462, D462) → E462Segment(C463, D463) → E463Segment(C464, D464) → E464Segment(C465, D465) → E465Segment(C466, D466) → E466Segment(C467, D467) → E467Segment(C468, D468) → E468Segment(C469, D469) → E469Segment(C470, D470) → E470Segment(C471, D471) → E471Segment(C472, D472) → E472Segment(C473, D473) → E473Segment(C474, D474) → E474Segment(C475, D475) → E475Segment(C476, D476) → E476Segment(C477, D477) → E477Segment(C478, D478) → E478Segment(C479, D479) → E479Segment(C480, D480) → E480Segment(C481, D481) → E481Segment(C482, D482) → E482Segment(C483, D483) → E483Segment(C484, D484) → E484Segment(C485, D485) → E485Segment(C486, D486) → E486Segment(C487, D487) → E487Segment(C488, D488) → E488Segment(C489, D489) → E489Segment(C490, D490) → E490Segment(C491, D491) → E491Segment(C492, D492) → E492Segment(C493, D493) → E493Segment(C494, D494) → E494Segment(C495, D495) → E495Segment(C496, D496) → E496Segment(C497, D497) → E497Segment(C498, D498) → E498Segment(C499, D499) → E499Segment(C500, D500) → E500Segment(C501, D501) → E501mean(L2:L500) → cmean(J2:J1000) → bDistance(f, g) → dPoint(f) → GOrthogonalLine(G, f) → mIntersect(m, g) → HDistance(G, H) → distanceGHVector(G, H) → uName(invprop) → J1
AI Drawing Prompt Reference
Draw a set of parallel lines to represent evenly spaced “fences”. For many repeated trials, generate a “needle” by choosing a random center position and a random direction angle, with a fixed needle length, then draw a line segment along that direction. For each trial, check whether the needle intersects the parallel lines (use the intersection points as evidence of crossing) and count the crossing for that trial. Accumulate these counts over many trials and compute the mean frequency of crossings to approximate the crossing probability, illustrating the link between geometric probability and an empirical estimate of π. The scene should show the parallel lines and the repeated randomly placed needle segments driving the running statistics.
The prompt above was compiled from the original drawing commands and is for reference only.