Exploring Marjorie Rice’s Tessellations: Art, Math, and Geometry

Gabrielle Birchak/ March 4, 2025/ Late Modern History, Modern History, Uncategorized

As we ven­ture into Women’s His­to­ry Month, I want­ed to talk about one of my favorite female math­e­mati­cians who didn’t even know she was a math­e­mati­cian, Mar­jorie Rice. 

Imag­ine a woman sit­ting at her kitchen table, sur­round­ed by stacks of scrap paper. A qui­et hum of every­day life swirls around her. Still, she’s deep in thought, her pen­cil trac­ing intri­cate pen­tagons over and over. It seems like a sim­ple pas­time, but in real­i­ty, she is rewrit­ing math­e­mat­i­cal his­to­ry. At that moment, Mar­jorie Rice, a self-taught math­e­mati­cian with no for­mal train­ing, was about to change a field that had long been dom­i­nat­ed by the most pres­ti­gious uni­ver­si­ties and schol­ars. She saw pat­terns where oth­ers saw impos­si­bil­i­ty, and her dis­cov­er­ies would soon chal­lenge the very foun­da­tion of geo­met­ric symmetry.

In today’s pod­cast, I uncov­er the sto­ry of Rice, who, with noth­ing more than paper, pen­cil, and an insa­tiable curios­i­ty, found her­self at the cen­ter of one of the most fas­ci­nat­ing break­throughs in math­e­mat­ics. Who was Mar­jorie Rice, and how did she defy all expec­ta­tions to make her mark on the world of geom­e­try? Let’s find out.

Mar­jorie Rice, nee Jeuck, was born in Flori­da in 1923. She grew up in an era when oppor­tu­ni­ties for women in math­e­mat­ics were scarce. She was intel­li­gent for her age. She stud­ied in a one-room school but skipped two grades and was allowed to study with old­er stu­dents. She was nev­er able to attend col­lege because her fam­i­ly couldn’t afford it. She mar­ried in 1945 to a gen­tle­man named Gilbert Rice. They left Flori­da for Wash­ing­ton, DC, where Gilbert worked in a mil­i­tary hos­pi­tal. To bring in extra mon­ey for the young cou­ple, Rice worked as a com­mer­cial artist briefly until they moved to San Diego, CA, with their infant son, who sad­ly died. She went on to have five oth­er chil­dren[1].

While she showed an apti­tude for num­bers, her for­mal edu­ca­tion nev­er extend­ed beyond high school. Instead of pur­su­ing high­er edu­ca­tion, she mar­ried young, altru­is­ti­cal­ly ded­i­cat­ing her­self to fam­i­ly life as a wife and moth­er. Her days were filled with the rou­tines of domes­tic life that includ­ed car­ing for her chil­dren, run­ning a house­hold, and man­ag­ing the count­less respon­si­bil­i­ties that came with it. Being a mom is tru­ly a full-time job. Math­e­mat­ics, at least aca­d­e­m­i­cal­ly, was far from her world.

She had an inquis­i­tive mind. In the 1970s, Rice’s math­e­mat­i­cal jour­ney began amid the every­day rhythm of her life. At the time, she had no rea­son to believe she would ever con­tribute to math­e­mat­ics, let alone make a dis­cov­ery that would stand along­side the work of pro­fes­sion­al math­e­mati­cians. But a sim­ple spark of curios­i­ty changed everything.

She had pur­chased a sub­scrip­tion to Sci­en­tif­ic Amer­i­can. The pur­chase was under the guise of a gift for her son. But while her kids were at school, she would pore over the mag­a­zine and specif­i­cal­ly looked for­ward to read­ing Mar­tin Gardner’s col­umn titled “Math­e­mat­i­cal Games.”

Mar­tin Gard­ner was a renowned sci­ence writer and pop­u­lar­iz­er of recre­ation­al math­e­mat­ics. His col­umn cap­ti­vat­ed read­ers by pre­sent­ing fas­ci­nat­ing math­e­mat­i­cal puz­zles and prob­lems that were acces­si­ble yet deeply com­plex. His work inspired gen­er­a­tions of math­e­mati­cians, logi­cians, and puz­zle enthusiasts.

In 1975, Rice came across one of Gardner’s arti­cles that focused on the tiling of the plane using con­vex pen­tagons. This top­ic has fas­ci­nat­ed math­e­mati­cians for decades because while squares, tri­an­gles, and hexa­gons can eas­i­ly tile the plane, mean­ing they can cov­er a flat sur­face with­out gaps or over­laps, pen­tagons are trick­i­er. At the time, math­e­mati­cians believed they had iden­ti­fied all the types of pen­tagons that could tile the plane, and the con­sen­sus was that only five such class­es existed.

Rice found her­self cap­ti­vat­ed by this prob­lem. Some­thing about the idea of tiling, the way shapes fit togeth­er, and the hid­den order behind seem­ing­ly com­plex pat­terns sparked her imag­i­na­tion. She was not a math­e­mati­cian by trade but a keen observ­er and a puz­zle solver at heart. The idea that only five types of pen­tagons could tile the plane didn’t sit right with her. She felt there was more to the sto­ry, that some­thing had been overlooked.

With no for­mal train­ing beyond high school alge­bra, she eval­u­at­ed the prob­lem her­self. Most peo­ple might have been deterred by the lack of cre­den­tials, but Rice was not most peo­ple. She had a nat­ur­al sense of won­der and per­sis­tence that defied the lim­i­ta­tions of tra­di­tion­al edu­ca­tion. Armed with a pen­cil, some paper, and an insa­tiable curios­i­ty, she began sketch­ing pen­tagons on scraps of paper at her kitchen table, explor­ing their poten­tial to tessellate.

She devised her own nota­tion sys­tem to keep track of her find­ings, using sym­bols and meth­ods entire­ly of her own mak­ing. She wasn’t bur­dened by the con­ven­tions of for­mal math­e­mat­i­cal train­ing. In some ways, her lack of con­ven­tions was an advan­tage. She approached the prob­lem with a fresh per­spec­tive, unen­cum­bered by assump­tions. She would sketch pen­tagons, test their arrange­ments, and search for pat­terns for hours. What start­ed as a puz­zle of inter­est soon became a qui­et obsession.

Rice’s intu­ition led her to ques­tion what the experts had accept­ed as fact. Could it real­ly be that only five types of con­vex pen­tagons could tile the plane? As she worked, she began uncov­er­ing new pos­si­bil­i­ties, pen­tagons that no one had pre­vi­ous­ly clas­si­fied. With every dis­cov­ery, her excite­ment grew. She wasn’t just play­ing with pat­terns; she was push­ing the bound­aries of math­e­mat­i­cal knowledge.

Rice, a valu­able and devot­ed home­mak­er with no advanced edu­ca­tion in math­e­mat­ics, was about to chal­lenge an estab­lished belief in the math­e­mat­i­cal world. And she was doing it from her kitchen table, immers­ing her­self in the world of math­e­mat­ics and dis­cov­er­ing a love for learn­ing that would give her a men­tal reprieve from mak­ing school lunch­es and clean­ing the house. And as a mom and a math­e­mati­cian, I get it! Math is by far the most extra­or­di­nary escape!

Rice’s inquis­i­tive mind was not con­fined by degrees or for­mal train­ing, and she would soon ven­ture into an extra­or­di­nary dis­cov­ery. Her jour­ney was only beginning.

Recognition and Collaboration

By the late 1970s, Rice had made remark­able progress in her study of pen­tag­o­nal tiling. Through sheer deter­mi­na­tion and inge­nu­ity, she dis­cov­ered four new types of pen­tag­o­nal tilings, which was an aston­ish­ing feat con­sid­er­ing that, pri­or to her work, math­e­mati­cians believed they had iden­ti­fied all pos­si­ble types. Excit­ed about her rev­e­la­tions, she wrote to Mar­tin Gard­ner to show him her dis­cov­er­ies. Gard­ner then for­ward­ed her work to math­e­mat­ics Pro­fes­sor, Esch­er Schol­ar, and tes­sel­la­tion expert Doris Schattschnei­der, who would help val­i­date and ampli­fy Rice’s contributions.

At first, Schattschnei­der was skep­ti­cal; after all, Rice was an ama­teur with no for­mal math­e­mat­i­cal train­ing. But when she exam­ined Rice’s care­ful, sys­tem­at­ic approach and pre­cise geo­met­ric nota­tion, she real­ized that Rice had indeed uncov­ered new tiling pat­terns that had elud­ed pro­fes­sion­al math­e­mati­cians. Schattschnei­der took it upon her­self to ver­i­fy Rice’s find­ings and present them to the aca­d­e­m­ic world.

Rice’s first intro­duc­tion to the aca­d­e­m­ic world was through Schattschneier’s first arti­cle on her titled “In Praise of Ama­teurs” in Math­e­mat­ics Mag­a­zine in 1977.[2] In Schattschneider’s sec­ond arti­cle in 1978, titled “Tiling The Plane With Con­gru­ent Pen­tagons,” co-authored with Richard Guy and pub­lished in Math­e­mat­ics Mag­a­zine, Schattschnei­der not only con­firmed the valid­i­ty of Rice’s work but also expand­ed the known clas­si­fi­ca­tion of pen­tag­o­nal tilings sig­nif­i­cant­ly.[3] as a result, this defied the sci­en­tif­ic com­mu­ni­ties stance that no new tilings could be cre­at­ed. Rice’s work increased the num­ber of known con­vex pen­tag­o­nal tilings from eight to twelve. Her work altered the future stud­ies of tes­sel­la­tion stud­ies.[4]

Beyond the math­e­mat­i­cal sig­nif­i­cance, the recog­ni­tion of Rice’s work had a broad­er impact on the per­cep­tion of ama­teur con­tri­bu­tions to math­e­mat­ics. His­tor­i­cal­ly, math­e­mat­ics was a field reserved for those with advanced degrees and for­mal train­ing. Rice’s suc­cess chal­lenged this notion, prov­ing that inde­pen­dent researchers with enough curios­i­ty and per­se­ver­ance could make sub­stan­tial con­tri­bu­tions. Her sto­ry became an inspir­ing exam­ple of how intel­lec­tu­al dis­cov­ery is not lim­it­ed to the halls of academia.

Rice and Schattschnei­der con­tin­ued to col­lab­o­rate in numer­ous ways. Schattschnei­der helped refine Rice’s nota­tion and ensured her work could be under­stood with­in the broad­er math­e­mat­i­cal discourse.

The lega­cy of Mar­jorie Rice extends beyond the pen­tagons she dis­cov­ered. Her work inspired oth­ers to look more close­ly at tiling prob­lems and con­tributed to a renewed inter­est in the role of ama­teurs in math­e­mat­i­cal research. Today, her pen­tag­o­nal tilings are stud­ied in math­e­mat­i­cal cur­ric­u­la and remain essen­tial to tes­sel­la­tion the­o­ry. Her journey—one that began with a sin­gle Sci­en­tif­ic Amer­i­can article—demonstrates the pow­er of curios­i­ty and the unex­pect­ed places it can lead.

Math­e­mat­i­cal Meth­ods of Mar­jorie Rice

Rice’s math­e­mat­i­cal process was metic­u­lous and inven­tive, built upon a com­bi­na­tion of geo­met­ric rea­son­ing, pat­tern recog­ni­tion, and self-taught meth­ods. Lack­ing for­mal aca­d­e­m­ic train­ing in high­er math­e­mat­ics, she devel­oped her own nota­tion sys­tem. She used log­i­cal deduc­tion to ana­lyze pen­tag­o­nal tilings.

One of Rice’s key strate­gies was geo­met­ric rea­son­ing. She sys­tem­at­i­cal­ly exam­ined pen­tag­o­nal prop­er­ties, iden­ti­fy­ing rela­tion­ships between angles and side lengths that allow seam­less tiling. Unlike trained math­e­mati­cians who might approach the prob­lem with estab­lished for­mu­las, Rice used a hands-on approach, sketch­ing pen­tagons by hand and test­ing their fit.

To orga­nize her find­ings, she devel­oped her own cus­tom nota­tion sys­tem. This nota­tion allowed her to track angle rela­tion­ships and tiling pat­terns effi­cient­ly. With­out the for­mal lan­guage of topol­o­gy or advanced geom­e­try, she cre­at­ed her own short­hand for describ­ing and clas­si­fy­ing pen­tagons based on their abil­i­ty to tessellate.

Rice also relied on basic alge­bra and trigonom­e­try. She cal­cu­lat­ed angles and ana­lyzed the geo­met­ric prop­er­ties of pen­tagons, ensur­ing that new types she dis­cov­ered could fit with­in known tes­sel­la­tion pat­terns. Though she did not use advanced math­e­mat­i­cal tools, her log­i­cal approach par­al­leled the method­olo­gies used by pro­fes­sion­al mathematicians.

Anoth­er crit­i­cal aspect of her process was pat­tern recog­ni­tion and iter­a­tion. She worked tire­less­ly, test­ing dif­fer­ent pen­tag­o­nal shapes and refin­ing her approach through repeat­ed tri­al and error. This per­sis­tence led her to dis­cov­er four new types of pen­tag­o­nal tilings, some­thing that had elud­ed trained math­e­mati­cians for decades.

Rice’s meth­ods exem­pli­fy the pow­er of inde­pen­dent math­e­mat­i­cal inquiry. Her dis­cov­er­ies, val­i­dat­ed by pro­fes­sion­als like Doris Schattschnei­der, reshaped the under­stand­ing of pen­tag­o­nal tiling and demon­strat­ed that sig­nif­i­cant math­e­mat­i­cal con­tri­bu­tions can come from unex­pect­ed places.

Broad­er Impact and the Role of Women in Mathematics

Mar­jorie Rice’s sto­ry high­lights the impor­tance of diverse per­spec­tives in math­e­mat­ics. As a woman with­out for­mal train­ing, her con­tri­bu­tions defied expec­ta­tions and demon­strat­ed that math­e­mat­i­cal break­throughs can emerge out­side the aca­d­e­m­ic world. Her suc­cess is a tes­ta­ment to the val­ue of inde­pen­dent explo­ration and the impact of intel­lec­tu­al curiosity.

Through­out his­to­ry, many women have made pro­found con­tri­bu­tions to math­e­mat­ics, often teach­ing them­selves or over­com­ing sig­nif­i­cant bar­ri­ers to do so. Mary Ever­est Boole, a self-taught math­e­mati­cian, trans­formed math­e­mat­i­cal edu­ca­tion by pio­neer­ing visu­al­iza­tion and hands-on learn­ing tech­niques for chil­dren. Flo­rence Nightin­gale, wide­ly known as a nurse, was also a sta­tis­ti­cian who rev­o­lu­tion­ized data visu­al­iza­tion, using her inno­v­a­tive pie charts to influ­ence health­care poli­cies. Kather­ine John­son, though for­mal­ly trained, faced sys­temic bar­ri­ers as an African Amer­i­can woman but went on to make cru­cial cal­cu­la­tions for NASA’s ear­ly space mis­sions. Sim­i­lar­ly, despite her for­mal edu­ca­tion, Joan Clarke was fre­quent­ly over­looked due to her gen­der. Still, she played an essen­tial role in break­ing the Ger­man Enig­ma code dur­ing World War II. These women defied soci­etal norms and left a last­ing impact on math­e­mat­ics and beyond.

Math­e­mat­ics is not reserved for the priv­i­leged few who receive for­mal edu­ca­tion. Math belongs to any­one with curios­i­ty, deter­mi­na­tion, and a desire to under­stand the world. Whether through self-teach­ing, uncon­ven­tion­al learn­ing meth­ods, or sheer per­sis­tence, any­one can unlock the beau­ty and pow­er of math­e­mat­ics. The sto­ries of Mary Ever­est Boole, Flo­rence Nightin­gale, Kather­ine John­son, and Joan Clarke remind us that bar­ri­ers are meant to be bro­ken and that the pur­suit of knowl­edge knows no bounds. If you have ever doubt­ed your abil­i­ty to engage with math­e­mat­ics, let their lega­cies inspire you to explore, ques­tion, and dis­cov­er. The lan­guage of num­bers, pat­terns, and log­ic is wait­ing for you to make your mark.

Pen­tag­o­nal Tilings in the World Today

Mar­jorie Rice’s sto­ry is a tes­ta­ment to the bound­less nature of human curios­i­ty. With­out for­mal train­ing, she reshaped the land­scape of math­e­mat­ics, prov­ing that dis­cov­ery isn’t con­fined to ivory tow­ers or advanced degrees. Dis­cov­er belongs to any­one with pas­sion, per­se­ver­ance, and a will­ing­ness to ques­tion the unknown. Her work in pen­tag­o­nal tiling con­tin­ues to influ­ence art, archi­tec­ture, and sci­ence, remind­ing us that math­e­mat­i­cal beau­ty isn’t just found in text­books. It can also be found in the world around us, which includes intri­cate tile mosaics and hid­den pat­terns of nature, like a pineap­ple! If you didn’t see that video, be sure to vis­it my Fibonac­ci post on Insta­gram about math in a pineapple!

As we cel­e­brate Women’s His­to­ry Month, let’s rec­og­nize the impact of women like Mar­jorie Rice, who defied expec­ta­tions and expand­ed the fron­tiers of knowl­edge. Rice’s con­tri­bu­tions chal­lenged the sta­tus quo and reshaped our under­stand­ing of math­e­mat­i­cal possibilities.

How­ev­er, per­haps the most impor­tant les­son from Mar­jorie Rice’s jour­ney is this: Nev­er let self-doubt silence your curios­i­ty. She could have con­vinced her­self she wasn’t qualified—that she was “just” a home­mak­er with­out for­mal train­ing. She could have ignored the spark of inter­est that led her to uncov­er new math­e­mat­i­cal truths. But she didn’t. She ques­tioned, she explored, and she discovered.

If you’ve ever felt like an imposter, like you don’t belong in a par­tic­u­lar space or con­ver­sa­tion, remem­ber this: break­throughs don’t always come from experts. Some­times, they come from those bold enough to ask, “What if?” The next great dis­cov­ery might not come from a uni­ver­si­ty lab but from a kitchen table, a sketch­book, or a mind will­ing to see pat­terns where oth­ers see only chaos.

So, to all the hid­den math­e­mati­cians, artists, sci­en­tists, and dream­ers out there, keep ques­tion­ing, keep explor­ing, and, just like Mar­jorie Rice, nev­er stop tiling your own path to discovery.

Resources

Doris Schattschnei­der — The Sto­ry of Mar­jorie Rice — G4G13 Apr 2018, 2019. https://www.youtube.com/watch?v=q2gs9LpICCk.

Grün­baum, Branko, and G. C. (Geof­frey Col­in) Shep­hard. Tilings and Pat­terns. New York : W.H. Free­man, 1987. http://archive.org/details/isbn_0716711931.

Schattschnei­der, Doris. “In Praise of Ama­teurs.” Math­e­mat­ics Mag­a­zine 50, no. 4 (1977): 211–17.

———. “Tiling the Plane with Con­gru­ent Pen­tagons.” Math­e­mat­ics Mag­a­zine 51, no. 1 (Jan­u­ary 1, 1978): 29–44. https://doi.org/10.1080/0025570X.1978.11976672.

Wol­chover, Natal­ie. “Mar­jorie Rice’s Secret Pen­tagons.” Quan­ta Mag­a­zine, July 11, 2017. https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/.

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[1] Wol­chover, Natal­ie. “Mar­jorie Rice’s Secret Pen­tagons.” Quan­ta Mag­a­zine, July 11, 2017. https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/.

[2] Schattschnei­der, Doris. “In Praise of Ama­teurs.” Math­e­mat­ics Mag­a­zine 50, no. 4 (1977): 211–17.

[3] Schattschnei­der, Doris. “Tiling the Plane with Con­gru­ent Pen­tagons.” Math­e­mat­ics Mag­a­zine 51, no. 1 (Jan­u­ary 1, 1978): 29–44. https://doi.org/10.1080/0025570X.1978.11976672.

[4] Grün­baum, Branko, and G. C. (Geof­frey Col­in) Shep­hard. Tilings and Pat­terns. New York : W.H. Free­man, 1987. http://archive.org/details/isbn_0716711931.