Sec­tion 3 explores the social struc­ture of Flat­land through both its geom­e­try and strict hier­ar­chy. Build­ings in pop­u­lat­ed areas must fol­low a legal stan­dard that ensures safe­ty by lim­it­ing sharp angles, with pen­tag­o­nal forms being the low­est accept­able design. This reflects a broad­er cul­tur­al shift, where even archi­tec­ture mir­rors the dri­ve toward sym­me­try and refine­ment. Only in remote, unde­vel­oped regions might a square house still be found—an archi­tec­tur­al rel­ic seen more as a curios­i­ty than a func­tion­al space. Inhab­i­tants them­selves fol­low sim­i­lar rules of pro­gres­sion, where one’s num­ber of sides deter­mines social class. A Cir­cle stands at the top, near­ly indis­tin­guish­able from a per­fect curve, while Isosce­les Triangles—barely dis­tin­guish­able from straight lines—occupy the low­est ranks. Each class is shaped not just phys­i­cal­ly but by law, tra­di­tion, and expec­ta­tion.

A full-grown fig­ure in Flat­land typ­i­cal­ly mea­sures eleven inch­es across, rarely exceed­ing twelve. Women are straight lines, while sol­diers and labor­ers are Isosce­les Tri­an­gles with dan­ger­ous­ly sharp ver­tices. The small­er the base, the more point­ed and threat­en­ing their form, mak­ing some near­ly indis­tin­guish­able from Women. Among the mid­dle class are Equi­lat­er­al Tri­an­gles, whose three equal sides give them a more bal­anced appear­ance and more sta­ble place in soci­ety. Above them are the pro­fes­sion­al class—Squares and Pentagons—followed by the Nobil­i­ty, which includes increas­ing­ly many-sided fig­ures. As the num­ber of sides ris­es, so does social rank, with the most elite—those indis­tin­guish­able from Circles—occupying the sacred priest­ly class. This geo­met­ric sys­tem doesn’t just gov­ern appear­ance; it defines intel­li­gence, sta­tus, and des­tiny. More sides imply more sta­bil­i­ty, intel­li­gence, and con­trol, mak­ing form an absolute mea­sure of worth.

A unique law dic­tates that each male child should pos­sess one more side than his father, the­o­ret­i­cal­ly allow­ing soci­ety to evolve through gen­er­a­tions. A Square might father a Pen­ta­gon, and a Pen­ta­gon could even­tu­al­ly sire a Hexa­gon. This upward mobil­i­ty, how­ev­er, does not extend even­ly across all class­es. Sol­diers and laborers—Isosceles by form—do not typ­i­cal­ly pro­duce chil­dren with more than three sides. Their shapes are fixed by both biol­o­gy and sta­tus. Yet there remains a sliv­er of hope. Through long-term ser­vice, dis­ci­pline, and selec­tive inter­mar­riage arranged by the rul­ing Cir­cles, it is pos­si­ble to improve geo­met­ric form grad­u­al­ly. Slight mod­i­fi­ca­tions to the base and angle may yield more bal­anced off­spring after sev­er­al gen­er­a­tions.

Although rare, a true Equi­lat­er­al Tri­an­gle born from Isosce­les par­ents is treat­ed as a soci­etal tri­umph. Such a birth is seen as a prod­uct of inten­tion­al dis­ci­pline and intel­lec­tu­al advance­ment over many gen­er­a­tions. It demands not just bet­ter mar­riage arrange­ments but the steady devel­op­ment of intel­li­gence and restraint among ances­tors. When such a child is born, the event is cel­e­brat­ed across entire regions. After pass­ing rig­or­ous inspec­tions by the Social and San­i­tary Board, the child is con­firmed as a gen­uine Equi­lat­er­al. With rit­u­al cer­e­mo­ny, he is induct­ed into the class of Equi­lat­er­als and adopt­ed by child­less par­ents from that group. His bio­log­i­cal par­ents, although hon­ored, must relin­quish him. Their sac­ri­fice rein­forces the belief that advance­ment belongs not to indi­vid­u­als but to the sys­tem itself.

This gen­er­a­tional pro­gres­sion reflects Flatland’s unyield­ing com­mit­ment to order and form. The gov­ern­ment rewards those who align with the sys­tem and pun­ish­es or excludes those who devi­ate. Yet, the sys­tem itself ensures that progress is slow and cal­cu­lat­ed. Advance­ment is pos­si­ble, but it is designed to be dif­fi­cult and earned over time. The rigid struc­ture pre­vents chaos, but it also lim­its spon­tane­ity and inno­va­tion. Intel­li­gence is tied to shape, and shape is inher­it­ed or improved only under strict con­trol. In Flat­land, mobil­i­ty isn’t deter­mined by mer­it alone—it’s a pre­cise result of geom­e­try, blood­line, and obe­di­ence. The ulti­mate reward is absorp­tion into the priest­ly class, where indi­vid­u­al­i­ty dis­ap­pears into per­fec­tion. It’s a soci­ety where everything—status, suc­cess, even parenthood—is mea­sured in sides.