2d:null 2e:["$","title",null,{"children":"Mayan numerals \n DRAFT | Toph Tucker | Observable"}] 33:Ta07,md`# Mayan numerals ${draft()} ##### 36 bc – 17th century ad ∙ [Wikipedia](https://en.wikipedia.org/wiki/Maya_numerals) ∙ [MacTutor](http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html) --- On July 12, 1562, the Franciscan friar Diego de Landa, on a mission in the city of Izamal in Yucatán to convert the Mayan people to Catholicism, gathered up and burned every instance of their writing he could find — around twenty seven books and five thousand images — thinking them the work of diabolical forces. ${quote( "gates", 0, "bottom" )} Recalled to Spain and chastised for his zeal, de Landa wrote a book as a sort of penance, to which we now owe most of what we know of what he destroyed. But his manuscript was lost, and the last Mayan city fell in 1697. None of their writing was known to survive; it was soon doubted that they’d ever had the ability. In 1739, Johann Christian Götze, director of the Royal Library at Dresden, purchased 78 pages of bark paper marked in vegetable dyes from a private collection in Vienna. There at the library it remained an obscure curiosity for nearly a century until in 1828, Constantine Samuel Rafinesque noticed similarities to etchings at the Maya ruins in Palenque, Mexico, establishing it as a surviving Mayan book, soon to be known as the Dresden codex. ${quote( "dresden" )} Champollion had just cracked the Rosetta stone in Paris, thus finally deciphering the ancient Egyptian hieroglyphs, and Rafinesque was eager to follow with the Mayan codex, but his attempts yielded gibberish. Only one thing was obviously familiar and intelligible. Here, you try:

“This page of Demotic has letters and numbers,” he wrote, “these represented by strokes meaning 5 and dots meaning unities as the dots never exceed 4.” ${quote( "rafinesque", 0 )} Thus number itself was his Rosetta stone, providing a familiar toehold in the foreign script. But from there the road was slow and hard. Rafinesque died in 1940, having mode no more progress. At just that time, the travel writer John Lloyd Stephens was exploring the Yucatán with topographical artist Frederick Catherwood; from 1841 to 1844, their narratives, and Catherwood’s meticulous engravings, brought widespread attention to the lost civilization. ${quote( "catherwood", 0 )}`34:T5a6,md`In 1862, after exploring Mesoamerica for himself, Charles Étienne Brasseur de Bourbourg discovered an abridged copy of de Landa’s manuscript in Madrid. established the days and months, massive breakthrough. but bourbourg's analyses of de landa and his subsequent discovery of a second codex… two years later, he published a reproduction, with a French translation. Another two years later, he identified a second codex in a private collection. His discoveries were massive, but his analyses advanced nothing.${quote( "coe1992", 0 )} But the key recorded by de Landa allowed Ernst Förstemann, in a series of publications from 1880 up to his death in 1906, to make a series of breakthroughs. He found (It is hard to grasp today, so far into the age of mechanical reproduction, what a challenge and labor it was to let scholars _see_ something!) The decipherment of the numerals and dates ran so far ahead of the rest that for a long time the dominant school of thought was that numerals and dates were essentially all that the writing contained. (Surely we, too, will be remembered only for what our successors can most clearly understand of our works. Maybe it’ll be our spreadsheets, our microprocessor architectures, our formulations of tempered and chemically strengthened glass, or Unicode.) The story of decipherment continues, but here our understanding of the numerals at least reaches its modern state, so let us pause.`35:T4df,getNumeral = (n, options) => { if (typeof options !== "object") options = {}; const { width = 40, height = 32, stroke = "black", fill = "none" } = options; const ratio = 40 / 32; const w = Math.min(width, height * ratio); const h = Math.min(height, width / ratio); const dh = h / 8; const nFives = ~~(n / 5); const nOnes = Math.floor(n % 5); const fives = d3.range(nFives).map( i => svg`` ); const ones = d3.range(nOnes).map( i => svg`` ); const zeros = ~~n === 0 ? svg`` : ""; return svg``; }36:T535,md`As in many numeral systems — though not the one we use today — there is an agglomerative principle at work whereby a mark repeated stands for that many times the value of that mark; then, at some point, multiple marks are collapsed into a single mark that stands for all of them. Concatenation implies addition, to a point. These concatenative-agglomerative systems (vs hashed-positional) tend to be like: there's a bijection between elements and marks, where more marks denote a higher number, and at various intervals you clump them into a single mark that stands for a set number. Which is very intuitive, because it's basically how objects in the world work (HIERARCHY OF RELATIVELY INDEPENDENT HIGHLY INTEGRATED). As opposed to the sort of HASHING of simple-arbitrary-positional (adjacencies are meaningless / uninvertible). - Cheap to increment without erasing, at least four times out of five! Sounds valuable on clay! - What's your loss function? Do you want adjacent numbers to look similar or not? - "Hashed" could also be called "alphabetical"… - Agglomerative feels idiomatic, ergonomic. Decimal numerals feel comparatively mechanical and harsh (metric)! Sorry pal, I don’t care how “nice” and “round” a number it is, it gets as many digits as the rest. - Greek feels like worst of both worlds, lol.`37:T840,mayanAnimation = { const duration = 450 const r = 10 const w = r * (subbase + 3) const h = r * 13 const digits = getDigits(n, base).reverse() const marks = d3.merge(d3.merge( digits .map(getMarks) .map((d, i, arr) => d.map(dd => d3.range(base ** (arr.length - i - 1)).map(() => dd))) )) const sel = d3.select(this || html``) let mark = sel.selectAll("path.mark").data(marks) mark.exit() .transition() .duration(duration) .attr("transform", "translate(500, 0)") .attr("opacity", 0) .remove() mark = mark.enter().append("path") .attr("d", d => poly(getSquircle(r, Number(d.type === "one")))) .attr("transform", (d, i) => `translate(500, ${d.type === "one" ? d.i * h + -r * 2 : d.i * h + r + r * 3 * (d.length - d.j - 1)})`) .attr("opacity", 0) .attr("class", "mark") .merge(mark) mark .transition() .duration(duration) .attr("opacity", 1) .attr("transform", d => d.type === "one" ? `translate(${d.length === 1 ? w / 2 : d.j * w / (d.length - 1)}, ${d.i * h + -r * 2})` : `translate(${r * 2 * d.k}, ${d.i * h + r + r * 3 * (d.length - d.j - 1)})`) .attr("d", d => poly(getSquircle(r, Number(d.type === "one")))) let zeroSel = sel.selectAll("path.zero") .data(digits.map((value, place) => ({place, value})).filter(d => !d.value)) zeroSel.exit() .transition() .duration(duration) .attr("opacity", 0) .remove() zeroSel = zeroSel.enter() .append("path") .attr("class", "zero") .attr("opacity", 0) .attr("d", zero(w)) .attr("transform", d => `translate(0, ${d.place * h})`) .merge(zeroSel) zeroSel .transition() .duration(duration) .attr("opacity", 1) .attr("transform", d => `translate(0, ${d.place * h})`) sel .transition() .duration(duration) .attr("viewBox", `0 ${-r * 4} ${w} ${digits.length * h}`) return sel.node() }38:T1404,md`How much blood was spilled because the number of Earth rotations in an orbit is not integrally divisible by the number of fingers and toes on the human body? Further progress was stymied by believing, with de Landa, that the Mayan script was alphabetic. (De Landa had commanded a Mayan to ${TK()}) Coe: nobody thought to consult the _present-day Mayan speakers!!!_ Just about the only reason any Mayan codices survived at all is that the Holy Roman Empire’s exfiltration of Cortés’s pillaging outran de Landa’s purging. In 1519, forty-three years before the friar’s _auto de fé_, Cortés shipped the “Royal Fifth” he owed to Charles V to Seville, including several books. From Seville they went to Valladolid, and then Brussels, and then the trail goes dark. The one we saw above showed up in Vienna in 1739; another in Paris in 1859; and another in Madrid in the 1860s. The one exception, the one that seems to have never left the New World, is a fourth found in a cave near Tortuguero, Mexico, in 1965. Its legitimacy is contested, largely on grounds of inconsistency with the other three. I am not an expert, but I don’t know how you could know how much consistency to expect based on a centuries-spanning civilization’s surviving corpus of three books. There is still only one sentence for which we have a definite translation. De Landa seems to have asked his translator — possibly Juan Nachi Kokom or Gaspar Antonio Chi, both Mayan nobles who worked with him — to say something. The Mayan wrote ${svg``}, pronounced it _ma in ka ti_, and translated it to “No quiero.”

“Broken Idol at Copán.” Frederick Catherwood, 1844

`39:T7e4f,sources = [ { key: "grolier", date: "c. 1100", alias: "Grolier codex (Maya Codex of Mexico)", urls: ["http://www.mesoweb.com/publications/Grolier/Grolier.pdf"], summary: "The fourth and latest Mayan manuscript to be discovered, now dated to be the oldest, and the only one still in the Americas. Found in a cave by looters in the 60s, bought by a Mexican collector, shown to Michael Coe, taken undeclared past U.S. Customs sandwiched in cardboard in a briefcase, exhibited at the Grolier Club in New York in 1971. The codex appears to show the synodic period of Venus. J. Eric S. Thompson decried it as a forgery (without seeing much of it) just before he died; before it was repatriated to Mexico, Coe ripped off a tiny corner of the paper for chemical testing, confirming its authenticity — for the first time of many, which kinda makes me wonder what it means to “confirm”.", links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Maya_Codex_of_Mexico" }, { label: "Grolier Club exhibit", url: "https://grolierclub.wordpress.com/2019/10/23/the-relationship-between-the-grolier-codex-and-the-grolier-club-of-new-york/" } ] }, { key: "dresden", date: "c. 1300", alias: "Dresden codex", urls: [ "https://en.wikipedia.org/wiki/File:DresdenCodexCorrectReadingSequence.jpg", "http://digital.slub-dresden.de/fileadmin/data/280742827/280742827_tif/jpegs/280742827.pdf" ], links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Dresden_Codex" }, { label: "Other codices", url: "https://en.wikipedia.org/wiki/Maya_codices" } ], summary: "The most famous of the surviving Mayan codices; 74 pages of colored vegetable dyes on “amate” bark paper. The history of its European study began when the director of the Royal Library at Dresden bought it from a private collector in Vienna in 1739. There the numerals were deciphered by Ernst Förstemann in 1894. Suffered water damage in a basement during the WWII bombings." }, { key: "madrid", date: "c. 1250–1450", alias: "Madrid codex", urls: ["http://www.famsi.org/mayawriting/codices/pdf/madrid_rosny_bb.pdf"], summary: "Discovered in Spain in pieces in the 1860s, partly by Charles Étienne Brasseur de Bourbourg. Seems to contain almanacs, horoscopes, and astronomical tables." }, { key: "paris", date: "c. 1450", alias: "Paris codex", urls: [ "http://gallica.bnf.fr/ark:/12148/btv1b8446947j.r=peresianus.langEN" ], summary: "Discovered in Paris in 1859 by Léon de Rosny. Seems to contain calendars and zodiacs." }, { key: "florentinecodex", date: 1545, authors: [ { name: "Bernardino de Sahagún", life: [1499, 1590], url: "https://en.wikipedia.org/wiki/Bernardino_de_Sahagún" } ], title: "Historia general de las cosas de nueva España", alias: "Florentine Codex", urls: ["https://www.wdl.org/en/item/10096/"], links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Florentine_Codex" } ], summary: "A massive, 12-book, 2,400-page compendium of life in Mesoamerica, compiled by the Franciscan friar, featuring more than 2,000 illustrations by native artists. Commonly known as the Florentine Codex, it deals with the Aztec, not the Maya, but it’s a giant incredible adjacent classic that I had never heard of." }, { key: "popolvuh", date: 1714, title: "Popol Vuh", language: "Spanish", authors: [ { name: "Francisco Ximénez", life: [1666, 1729], role: "trans." } ], urls: [ "https://library.osu.edu/projects/popolwuj/folios_eng/PWfolio_i_r_en.php", "http://www.mesoweb.com/publications/Christenson/PV-Literal.pdf" ], summary: "The Mayan creation epic, long passed down by oral tradition, somehow written down around 1550, was at this point translated to Spanish by the Dominican friar, to whom we owe its survival today.", links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Popol_Vuh" } ] }, { key: "landa", date: 1566, authors: [{ name: "Diego de Landa", life: [1524, 1579] }], language: "Spanish", title: "Relación de las cosas de Yucatán", urls: ["https://www.wayeb.org/download/resources/landa.pdf"], relations: [{ type: "translated in", key: "gates" }], summary: "Written upon his return to Spain, it is probably the single greatest account of the culture he did so much to erase." }, { key: "rafinesque", date: 1833, title: "Atlantic Journal and Friend of Knowledge", authors: [ { name: "Constantine Samuel Rafinesque", life: [1783, 1840], url: "https://en.wikipedia.org/wiki/Constantine_Samuel_Rafinesque" } ], urls: ["https://archive.org/details/atlanticjournal00rafigoog/"], summary: "Initial identification of the Dresden codex as Mayan.", quotes: [ [ 42, "But shall we be able to read these glyphs and inscriptions? without positively knowing in what language they were written! The attempt will be arduous, but is not impossible. In Egypt, the Coptic has been found such a close dialect of the Egyptian, that it has enabled you to read the oldest hieroglyphs. We find among the ancient dialects of Chiapa, Yucatan and Guatimala, the branches of the ancient speech of Otolum.… This page of Demotic has letters and numbers, these represented by strokes meaning 5 and dots meaning unities as the dots never exceed 4.… It might not be impossible to decypher some of these manuscripts written on metl paper: since they are written in languages yet spoken, and the writing was understand in Central America, as late as 200 years ago." ] ] }, { key: "stephens1841", date: 1841, title: "Incidents of Travel in Central America, Chiapas, and Yucatán", authors: [ { name: "John Lloyd Stephens", life: [1805, 1852] }, { name: "Frederick Catherwood", life: [1799, 1854] } ] }, { key: "stephens1843", date: 1843, title: "Incidents of Travel in Yucatán", authors: [ { name: "John Lloyd Stephens", life: [1805, 1852] }, { name: "Frederick Catherwood", life: [1799, 1854] } ] }, { key: "catherwood", date: 1844, urls: ["https://archive.org/details/gri_33125012602591/page/n4"], authors: [{ name: "Frederick Catherwood", life: [1799, 1854] }], title: "Views of ancient monuments in Central America, Chiapas and Yucatan", summary: "Featuring 25 plates by Catherwood, mostly drawn on his second trip to the Yucatan with Stephens (the dedicand of this volume), in 1941. Together with their previous narratives, contributed greatly to popular awareness of the Maya.", links: [ { label: "Illustrations", url: "https://www.oldbookillustrations.com/artists/catherwood-frederick/" } ], quotes: [ [ 1, "The monuments represented in this Volume seem, from their novelty and peculiar character, to demand some preliminary explanations of the circumstances under which they are found to exist, and the historic interest that attaches to them, as the most important aids we possess, for the investigation of that great unsettled problem — the origin of the inhabitants of the American continent, and the sources from whence their early civilisation was derived. No questions, merely antiquarian, have given rise to more earnest discussions than those involved in this subject; and, until of late years, the hardihood of the disputants has been in proportion to the scantiness of the evidence that had survived the ravages of conquest, and the iconoclastic bigotry of the earlier Christian missionaries. It is only within the present century that the attention of European scholars has been drawn to the fact, that a new and unexceptionable class of testimony, bearing directly on the Anti-Columbian History of the American continent, was within their reach; that there yet mouldered within the Forests of Yucatan and Guatemala, architectural and sculptural remains of vast size and mysterious purpose, still displaying (though yielding to a daily process of disintegration and decay) a high degree of constructive skill, and attesting, in their ornaments and proportions, to the prevalence of an indigenous and well established system of design, varying from any known models in the old world. The truth of this statement, though at first received with incredulity, has been satisfactorily established by later researches; and I may appeal to the following Drawings for its confirmation. They illustrate some of the more striking objects which engaged my notice as an Artist, during two expeditions, undertaken expressly with a view of exploring the ruined sites of Central America, and preserving some memorials of their present state. The first of these was devoted chiefly to the countries known under the above general title, including the States of Honduras, Guatemala, Chiapas, &c. The ruins at Copan and Palenque were visited during this journey, which occupied part of the years 1839 and 1840. A brief sojourn in Yucatan having shown the richness of the antiquarian harvest that there awaits the gleaner, a second journey, for its more thorough examination, was determined on, in the year 1841: in its progress most of the Drawings in the present Volume were made. The narrative of these expeditions will be found in the well-known works of Mr. J. L. Stephens, “Incidents of Travel in Central America, Chiapas, and Yucatan,” 4 vols. 8vo., New York and London; to them, and to the lately published “History of the Conquest of Mexico,” by Mr. Prescott, I must refer the reader desirous of further knowledge. In the one, he will meet with all the information that personal observation, directed by enterprise and enthusiasm, can supply; and in the other, all the light that a most extended range of research through the whole body of existing documentary evidence, can throw on the obscurity that shrouds the history of the unrecorded races — beyond the page of written annals — whose very existence we should be ignorant of, but for the contemplation of their colossal works, still before our eyes." ] ] }, { key: "bourbourg1864", date: 1864, title: "Relation des choses du Yucatán de Diego de Landa", authors: [ { name: "Charles Étienne Brasseur de Bourbourg", life: [1814, 1874] } ], summary: "Brasseur de Bourbourg discovered de Landa’s manuscript and printed a reproduction, including a translation into French.", relations: [{ type: "reproduces", key: "landa" }] }, { key: "bourbourg1869", date: 1869, authors: [ { name: "Charles Étienne Brasseur de Bourbourg", life: [1814, 1874] } ], title: "Manuscrit Troano : études sur le système graphique et la langue des Mayas", urls: ["https://archive.org/details/manuscrittroano01bras"], links: [ { label: "Wikipedia", url: "https://en.wikipedia.org/wiki/Charles_%C3%89tienne_Brasseur_de_Bourbourg" } ], summary: "Brasseur de Bourbourg identified what is now known as the Madrid codex in a private collection in 1866. His attempt to decipher it mostly failed, partly because, inspired by de Landa, he attempted to treat the glyphs as an alphabet. He seems to have decoded the numbers up to twenty, either oblivious to or ignoring that this had already been accomplished by Rafinesque. He recognized the importance of twenty, but did not establish the vigesimal place value system (i.e. the fact that a single dot could represent twenty), or the zero.", quotes: [ [ 126, "Les signes numéraux, autant qu’il est possible d’en juger par le premier coup d’œil, sont très-fréquents dans le _Manuscrit Troano_, aussi bien que dans le _Codex de Dresde_; mais les seuls dont on soit absolument certain sont les boules noires, employées généralement de un à dix, comme chez les Mexicains. Les boules rouges, qu’on y trouve alternativement avec les noires ou les blanches, ont-elles la même valeur? c’est probable, bien qu’elles aient une signification qui leur soit propre, en ce sens qu’elles se rapportent, selon toute apparence, à des choses ayant le feu pour objet; les noires elles-mêmes ont souvent une valeur locale en même temps que numérique. Je vais essayer d’éclaircir cette matière, autant qu’il me sera possible, dans le présent paragraphe; l’analyse des termes exprimant les principaux nombres mayas, comparés aux signes numéraux mexicains déjà connus, amènera, nous l’espérons, quelque solution satisfaisante pour ceux qui nous sont encore inconnus." ], [ 126, `1. • _Hun_, un, est le premier terme: on le trouve, en composition, dans quelques vocables, mais orthographié sans aspiration comme en français _un_. Il exprime, en maya aussi bien qu’en quiche, l’unité et la cohésion; mais il semblerait qu’à l’origine son acception fût celle d’aplanir, d’égaliser, bien que le vocable _un_ ou _hun_ exprimât encore l’idée que nous attachons à l’image du signe européen 1, c’est-à-dire le membre viril. _Hun_ est le nombre abstrait : lorsqu’il est joint à un nom quelconque, les Mayas disent: _humpel_, un. Ex.: _Humpel uinic_, un homme, c’est-à-dire un homme seulement. Le signe représentatif de ce nombre est, comme en mexicain, la boule noire •, qui paraît avoir été anciennement le signe d’un clou ou plutôt d’une cheville ou épine fixée dans une planche, par conséquent le 1 dans l’idée de la longueur partant d’une base plus grosse. Dans la langue nahualt, _ce_ ou _cen_ exprime le nombre _un_, et sa signification étymologique est celle d’un grain de maïs, _centli_, ou d’un petit grêlon, _cetl_, ce qui est comprimé, resserré, congelé, coagulé. En maya, le monosyllabe _cen_ qu’on prononce _qen_, est au fond identique au mexicain; car il a l’acception d’_un qui_, et joue le rôle d’un pronom interrogatif comme _quem_ en latin.` ], [ 129, "5. — _Ho_ ou _hopel_, cinq, est un vocable dont il serait difficile de déterminer l’étymologie: tout ce qu’on peut donner de certain à cet égard, c’est que _ho_ est le primitif d’un grand nombre d’autres vocables qui ont tous, plus ou moins, un sens faisant allusion à la ruine, à l’effondrement, à l’action violente de bouleverser. _Hoco_ ou _con_ est encore un verbe irrégulier qui signifie aller, le même que le _go_ de l’anglais. Le nombre cinq était-il représenté par cinq boules, comme le sont les nombres précédents par deux, par trois ou par quatre, c'est là la question qu’on s’adresse, en observant les pages du _Manuscrit Troano_. Nulle part, en effet, on n’y trouve de groupe, ayant l’apparence d’un signe numéral, qui contienne plus de quatre boules; le chiffre _cinq_ paraît constamment être désigné par la barre —, dont j’ai parlé plus haut, à propos de la surface du sol. Celle-ci s’exprime d’ordinaire par le mot _lah_, qui a l’acception d’un entier, d’un terme complet, tel qu’est le nombre _cinq_; il signifie ce qui est plat, étendu comme la main aux doigts étendus, faisant cinq. Quant aux chiffres suivants, ils se représentent par l’addition des quatre premiers successivement jusqu’à neuf inclusivement. C’est là ce que le lecteur se trouvera à même de remarquer, en continuant. Je n’affirmerai pas, d’une manière absolue, que les Mexicains aient usé du même signe pour _cinq_. Gama n’en dit rien; mais il se trouve employé d’une manière identique à celle du _Manuscrit Troano_ et du _Codex de Dresde_, dans le _Manuscrit mexicain d’Oxford_ (Laud. B. 65, nunc 678), dans le _Manuscrit mexicain de Bologne_ et dans le _Manuscrit Fegérvary de Pesth_, reproduits dans la collection de Kingsborough. Dans la langue nahuatl, _cinq_ s’exprimait par le vocable _macuili_ ou _macueli_, littéralement «torsion ou courbe, pli de la main,» de _ma_, primitif de _maitl_, main, et de _cueloa_, plier, retourner. «Ce qui démontre clairement, dit à ce sujet l'astronóme Gama, qu'à l'origine on distinguait chaque unité en repliant un doigt, jusqu’à compléter le nombre cinq en fermant la main.» Ajoutons que le caractère […] _manik_ donne précisément l’image du nom de nombre mexicain, _macuilli_, cinq." ], [ 134, "De 10 à 20 exclusivement, les nombres se forment en maya par l’addition pure et simple des nombres d’un à neuf au nombre dix, à l’exception, toutefois, du chiffre 11 qui se compose de 4 plus 7, les quatre boules, _bul_ ou _bol_, et _uuc_, sept. En mexicain, le même calcul a lieu jusqu’à 14 inclusivement; mais _quinze_ est un vocable tout à fait différent des précédents." ], [ 135, "Ainsi, d’après ce qu’on vient de voir, la numération maya, de cinq en cinq jusqu’à dix et de dix à vingt, est décimale: si, dans la langue quichée, un homme fait, _hun-vinac_, exprime l’idée de 20, dans le maya, elle se rend par _hun-kal_, une fermeture, un barrage, une agrafe, ce qui se comprend aisément quand on a vécu de la vie des Américains indigènes et qu’on connaît leurs usages. En effet, anciennement les maisons au Mexique, comme encore aujourd’hui les chaumières véritablement indigènes, dans l’Amérique centrale , n’avaient point de portes: l’entrée en était barrée simplement à l’aide d’une natte ou d’une claie en osier ou en cannes, dont quatre bambous plus gros que les autres faisaient toute la force; ce sont les quatre barres du barrage [four bars] qui apparaissent si fréquemment dans les documents. Dans le _Manuscrit Fegérvary_, les barres réunies en fagots sont souvent bien plus nombreuses, ce qui s’explique, peut-être, par des nombres excédant 20. De même que _lah_, en maya, a le sens d’un entier, _kal_, vingt, a encore celui d’un total. En mexicain, _cem-pohualli_, c’est-à-dire un compte rond, exprimant le même chiffre, se rendait, d’après tous les auteurs, par un petit drapeau [flag symbol] _pan_ ou _pantli_; mais Gama assure que ce drapeau, alternativement partagé endeux, en trois ou en quatre [checkered flag], par la suppression d’un, de deux ou de trois de ses carrés, pouvait présenter tour à tour, soit le nombre vingt, soit le vingt moins un, deux ou trois carrés, valant chacun 5; ce qui nous ramène plus ou moins aux quatre barres des calculs mayas. Bien que les barres se trouvent fréquemment dans les documents mexicains cités plus haut, je n’ai, en revanche, découvert, ni dans le _Manuscrit Troano_, ni dans le _Manuscrit de Dresde_, aucun signe identique au drapeau mexicain représentant 20." ] ] }, { date: 1915, authors: [{ name: "Sylvanus Griswold Morley", life: [1883, 1948] }], title: "An Introduction to the Study of the Maya Hieroglyphs", urls: [ "https://en.wikisource.org/wiki/An_Introduction_to_the_Study_of_the_Maya_Hieroglyphs", "https://www.gutenberg.org/ebooks/43491" ] }, { key: "cajori", date: 1928, authors: [{ name: "Florian Cajori", life: [1859, 1930] }], title: "A History of Mathematical Notations", urls: ["https://archive.org/details/historyofmathema031756mbp"], quotes: [ [ "§5", "The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era. [We have Babylonian tablets from the 3rd millennium B.C. evincing the principle of position, and attestations of a zero c. 200 B.C., but] it was not used in calculation." ], [ "§68", "The Maya system was vigesimal, except in one step. That is, 20 units (_kins_, or ‘days’) make 1 unit of the next higher order (_uinals_, or 20 days), 18 _uinals_ make 1 unit of the third order (_tun_, or 360 days), 20 _tuns_ make 1 unit of the fourth order (_Katun_, or 7,200 days), 20 _Katuns_ make 1 unit of the fifth order (_cycle_, or 144,000 days), and finally 20 _cycles_ make 1 _great cycle_ of 2,880,000 days." ], [ "§68", "The zero is represented by a symbol that looks roughly like a half-closed eye." ], ["§68", "The largest number found in the codices is 12,489,781."] ] }, { key: "gates", date: 1937, authors: [ { name: "Diego de Landa", life: [1524, 1579] }, { name: "William E. Gates", life: [1863, 1940], role: "trans." } ], title: "Yucatan Before and After the Conquest: The Maya", relations: [{ type: "translates", key: "landa" }], urls: [ "https://www.globalgreyebooks.com/content/books/ebooks/yucatan-before-and-after-the-conquest.pdf" ], quotes: [ [ 135, "These people also used certain characters or letters, with which they wrote in their books about the antiquities and their sciences; with these, and with figures, and certain signs in the figures, they understood their matters, made them known, and taught them. We found a great number of books in these letters, and since they contained nothing but superstitions and falsehoods of the devil we burned them all, which they took most grievously, and which gave them great pain." ] ] }, { date: 1966, title: "The Maya", authors: [{ name: "Michael D. Coe", life: [1929, 2019] }], quotes: [ [ 182, `Otto Neugebauer, the historian of science, considers positional, or place-value, numeration as ‘one of the most fertile inventions of humanity’, comparable in a way with the invention of the alphabet. Instead of the clumsy, additive numbers used by the Roman and so many other cultures of the world, a few peoples have adopted ‘a system whereby the position of a number symbol determines its value and consequently a limited number of symbols suffices to express numbers, however large, without the need for repetitions or creation of higher new symbols’.` ], [ 190, `Few studies can have advanced so little with so much effort as the decipherment of the Maya script. This is not to say that a great deal cannot be understood, but there is a difference between unravelling a meaning for a sign, and matching it with a word in the Maya tongue. Progress has been most rapid on those glyphs which are of mainly calendrical or astronomical significance. For instance, by the mid nineteenth century the Abbe Brasseur de Bourbourg had discovered Landa’s _Relación_, from which he was able to recognize day glyphs and interpret the bar-and-dot numeration in the codices. It was quickly discovered that Maya writing was to be read in double columns from left to right, and top to bottom. At the turn of the century all of the following had been correctly deciphered by scholars in Europe and America: the zero and twenty signs, the world directions and the colours, Venus, the months (also in Landa), and the Long Count. In a remarkable collaboration between astronomers and epigraphers, the mysteries of the Lunar Series had been unveiled by the early 1930s. But after these intellectual triumphs, fewer and fewer successes were scored, leading a few pessimists to the quite unfounded claim that there is little else in these texts but calendrical and astronomical mumbo-jumbo.` ], [ 192, `Moving semantically, to express a concept not easily visualized, one would picture something related to it, for instance a representation of a fire to express the word ‘hot’; almost all of the world’s scripts have adopted this principle at some time in their evolution. In its purest form such a system could be called ideographic, and could be ‘read’ without reference to any particular language. Numbers such as our own arabic ones are also ideographs for which all sorts of people have separate terms; so were the bars and dots of the Mesoamericans.` ] ] }, { date: 1973, title: "The Maya Scribe and His World", authors: [{ name: "Michael D. Coe", life: [1929, 2019] }], urls: [ "http://www.mesoweb.com/publications/MayaScribe/MayaScribe.lores.pdf" ], summary: "Guide to the 1971 Grolier exhibit that debuted the Grolier codex", relations: [{ type: "Reproduces", key: "grolier" }] }, { date: 1989, title: "Maya Writing", urls: ["https://www.jstor.org/stable/24987365"], authors: [ { name: "David Stuart", life: [1965] }, { name: "Stephen D. Houston", life: [1958] } ], summary: "Great concise summary of decipherment so far, especially the dawn of the phonetic era with Knorozov, Berlin and Proskouriakoff, with illustrations of how one word can have both a logographic and syllabic representation.", quotes: [ [ null, "The glyphic system in which these early texts (and the subsequent ones) were written was not invented by the Maya. On the contrary, it seems they built on an ancient form that had been in use for centuries among advanced cultures to the west. Writing was present in what is now the state of Oaxaca by about 700 B.C., as is shown by the discovery there of a monument inscribed with early glyphs." ], [ null, "Whatever the preCise origins of Maya writing, by the beginning of the Classic period of Maya culture in about A.D. 250 hieroglyphs were in use at hundreds of sites. During the Classic period, which lasted until A.D. 900 and represents the zenith of Maya culture, countless stone monuments had hieroglyphic signs carved on their surfaces. Writing also appeared in books made of bark paper and covered with jaguar skin, on painted and molded ceramics and on other portable objects." ], [ null, "In the Postclassic period that followed the collapse, writing was still used on some stone monuments and architecture and in bark-paper books, of which only four survive. Indeed, in the last unconquered Maya enclave (in northern Guatemala), writing probably continued until the 17th century. Today, although the written tradition has been lost, four million speakers of two dozen Maya languages still inhabit the Maya homeland." ], [ null, "The efforts at deciphering Maya writing actually began soon after the Spanish conquest of 1520. Indeed, the first students of Maya hieroglyphs were Spanish friars who were attempting to convert the Maya to Christianity. The most famous of these was Diego de Landa, third bishop of Yucatan, who in 1566 compiled a treatise called _Relación de las casas de Yucatán_ (roughly, “An Account of the Things of Yucatán”)." ], [ null, "Although the rituals were significant — and were complemented by others in which the blood of captives was shed — it would be wrong to put too much emphasis on the bloodletting rites or to form the impression that Maya society was dominated by the shedding of blood." ], [ null, "Glyphs _a_ and _d_ spell the Maya word _ka-ka-w(a)_, from which the English “cacao” is derived." ] ] }, { key: "coe1992", date: 1992, title: "Breaking the Maya Code", authors: [{ name: "Michael D. Coe", life: [1929, 2019] }], summary: "Brimming with professional respect and awe and occasional disdain at the Mayanists who’ve come before; in reading it, one begins to understand how we have any sense at all of things that came so long before and were so long lost.", quotes: [ [ null, "Brasseur’s commentary accompanying the Troano facsimile is a case study in the fallacy of misplaced concreteness. Without the least idea of the order in which the Troano glyphs were to be read (he got them backwards), he began applying Landa’s ‘A,B,C,’ _as an alphabet_, to each glyph. The results were disastrous: his readings were nonsensical and patently false, although he remained oblivious to criticism. His incredible carelessness even led him to invent a letter for the Landa “alphabet” which is nowhere to be found in the original. As a result, any kind of phonetic approach to Maya hieroglyphic writing was thrown into an opprobrium from which it took almost a century to recover. Brasseur went off the deep end, and not just in his ill-founded application of the data from Landa. As historian Robert Brunhouse has told us, ‘… as book after book appeared, his ideas grew more strange and his explanations more attenuated, so that serious readers who had respected him increasingly lost confidence in his utterances. Why his fertile imagination got the better of him is not clear.’ Diffusionist obsessions seem to have been the trap of many otherwise sane Americanists, who just could not bring themselves to believe that New World civilizations were autochthonous. Remember the Lost Tribes of Israel hobbyhorse that bankrupted Kingsborough? Well, in Brasseur’s case his hobbyhorse was the myth of Atlantis, the continent that had supposedly sunk beneath the waves in ancient times, from which cataclysm refugees bearing the arts of civilized life had supposedly reached Yucatán and Central America." ] ] }, { date: 2001, title: "Reading the Maya Glyphs", authors: [ { name: "Michael D. Coe", life: [1929, 2019] }, { name: "Mark Van Stone" } ] }, { date: 2012, title: "The Calligraphic Zero", authors: [{ name: "David Stuart", life: [1965] }], urls: ["https://mayadecipherment.com/2012/06/15/the-calligraphic-zero/"], summary: "Describes variations of the zero glyph, from the Classic era to the Colonial, in stone inscriptions and codices." }, { date: 2012, title: "The Misunderstanding of Maya Math", type: "article", authors: [{ name: "David Stuart", life: [1965] }], urls: [ "https://mayadecipherment.com/2012/05/02/the-misunderstanding-of-maya-math/" ], summary: "An essential corrective to the basic Wikipedia narrative of Mayan numerals.", quotes: [ [ null, "As far as I am aware no purely vigesemal place-notation system was ever written this way. It’s true that in Mayan languages numbers are base-20 in their overall structure, just as in most Mesoamerican languages. In Colonial Yukatek, for example, we have familiar terms for these units: k’al (20), bak’ (400), pik (8,000), and so on. However, ancient scribes never represented these units in a columnar place notation system, as is so commonly described in the textbooks. That format was instead always reserved for a for the count of time, in what we know as the Long Count. That system is mostly vigesimal, but it is skewed in one of its units (the Tun, of 360 days) in order to conform as much as possible to the number of days in the solar year (365). To reiterate: the columns of numbers we find in the pages of the Dresden Codex or painted on the walls of Xultun (stay tuned, folks…) are all day counts; the positional notation system was never used for reckoning anything else. ❡ In the ancient inscriptions non-calendrical counts using large numbers are quite rare, mostly found in connection to tribute tallies, such as the counting of bundled cacao beans. But in those settings the scribes always seem to show nice rounded numbers (as in ho’ pik kakaw, “5×8,000 [40,000] cacao beans,” shown in the murals of Bonampak) without all the place units we know from the Long Count. In the Dresden and Madrid codices, counts of food offerings are given as groupings of WINIK (20) signs with accompanying bars and dots for 1-19. In this way a cluster of four such elements (4×20) with 19 writes 96 (See Love 1994:58-59; Stuart, in press)." ] ] } ]3a:[] 31:{"id":"d112bf2ea7f94a35","slug":"mayan-numerals","title":"Mayan numerals \n DRAFT\n ","thumbnail":"d6c75099871632e69e5ab1dacaeb8a46976826f6ad35f159d2ac4f01a94c768d","access_level":"public","create_time":"$D2019-07-07T23:13:10.339Z","update_time":"$D2022-09-19T21:11:29.412Z","view_time":null,"first_public_time":"$D2019-07-11T17:06:06.012Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"97290e936c339d8e","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":"pro_2024"},"version":1553,"publish_version":1553,"role":null,"authored":false,"authors":[{"type":"individual","id":"0bb2842b01e0c652","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":null}],"members":[{"type":"individual","id":"0bb2842b01e0c652","login":"tophtucker","name":"Toph Tucker","avatar_url":"https://avatars-next.observableusercontent.com/3706c31a8f9c59bb716c022c04b9c073144cd1368b78ac85f49139be6c93c34e","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"d40cc4f153f404b4","slug":"how-we-do-math","title":"How We Do Math","owner":{"id":"97290e936c339d8e","login":"tophtucker","name":"Toph Tucker"}},"type":"notebook","body":{"title":"Mayan numerals \n DRAFT\n ","cells":[{"id":564,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1024,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":328,"value":"html`

${d3\n .range(0, 20, 5)\n .map(d => html`

${d3.range(d, d + 5).map(getNumeral)}

`)}

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":100,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":0,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"md`Demonstration of agglomerative principle:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":472,"value":"viewof n = timerSlider(0, 400, 500)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":482,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":572,"value":"md`There was that whole nutty 2012 thing, which hinged on… ticking over from 12.19.19.17.19 to 13.0.0.0.0, which was when the previous cycle had supposedly ended. https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar\n\n> The pictures and glyphs were painted by skilled craftsmen using thin brushes and vegetable dyes.[8] Black and red were the main colors used for many of the pages.[9] Some pages have detailed backgrounds in shades of yellow, green, and the Mayan blue.[10] https://en.wikipedia.org/wiki/Dresden_Codex\n\nThe numerals were hollow when etched in stone (e.g. in [La Mojarra Stela 1](https://en.wikipedia.org/wiki/La_Mojarra_Stela_1)), or filled-in when done with dyes on bark paper (e.g. in the [Dresden codex](https://en.wikipedia.org/wiki/Dresden_Codex)).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":825,"value":"md`They were often ornamented, somewhat obscuring legibility. (from Morley “An Introduction to the Study of the Maya Hieroglyphs”)\n\n

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1250,"value":"md`### Calendar, long count`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1248,"value":"dateToLongCount(date).reverse()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1204,"value":"html`${dateToLongCount(date)\n .reverse()\n .map(getNumeral)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1182,"value":"viewof date = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":841,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":831,"value":"md`---\n\n#### Source chronology`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":760,"value":"bibliography(sources)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":469,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":758,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":110,"value":"html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":364,"value":"base = 20","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":367,"value":"subbase = 5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":155,"value":"zero = size => {\n // by Salgueiro & Sarang https://en.wikipedia.org/wiki/File:Mayan00.svg\n const path =\n \"M32 0a32 16 0 1 0 1 0zm-30 11q30 18 61 0m-42 -10q-7 8 0 18m12 -19q-8 10 0 20m11 -19q-7 8 0 18\";\n const naturalSize = 65;\n return scale(size / naturalSize)(path);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":396,"value":"getMarks = (n, i = 0) => {\n const nGroups = ~~(n / subbase)\n const nOnes = n % subbase\n const groups = d3.merge(d3.range(nGroups)\n .map(j => ({i, j, length: nGroups, type: \"group\"}))\n .map(d => d3.range(subbase).map(k => ({k, ...d}))))\n const ones = d3.range(nOnes)\n .map(j => ({i, j, length: nOnes, type: \"one\"}))\n return [...groups, ...ones]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"d3 = require(\"d3\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":270,"value":"import { scale } from \"@tophtucker/pathstuff\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":346,"value":"import {getSquircle} from \"@tophtucker/interpolate-between-square-and-circle\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":397,"value":"import {getDigits} from \"@tophtucker/babylonian-numerals\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":447,"value":"import {timerSlider} from \"@tophtucker/roman-numerals\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":511,"value":"import {poly, draft, bibliography, bibliographyStyles, quoter, TK} from \"@tophtucker/scrapbook\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1000,"value":"quote = quoter(sources)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1229,"value":"md`---\n\n#### Long Count`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1057,"value":"dateToJulianDay = date => +date / 86400000 + 2440587.5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1110,"value":"julianDayToDate = d => new Date((d - 2440587.5) * 86400000)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1064,"value":"// Debating 584285 vs. 584283: http://edj.net/mc2012/fap3.html\ncorrelationConstant = 584283","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1066,"value":"// big-endian\nlongCountToJulianDay = d =>\n d.reduce(\n (acc, cur, i) => acc + cur * product(longCountBases.slice(0, i)),\n 0\n ) + correlationConstant","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1128,"value":"julianDayToLongCount = d =>\n getDigitsVariableBase(d - correlationConstant, longCountBases)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1171,"value":"dateToLongCount = d => julianDayToLongCount(dateToJulianDay(d))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1140,"value":"// big-endian\ngetDigitsVariableBase = (n, bases) => {\n const digits = new Array(bases.length);\n const values = basesToValues(bases);\n let m = n;\n let i = bases.length - 1;\n while (m && i) {\n const d = ~~(m / values[i]);\n digits[i] = d;\n m -= d * values[i];\n i--;\n }\n digits[i] = m;\n return digits;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1192,"value":"dateFormat = d3.timeFormat(\"%Y-%m-%d\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1126,"value":"longCountToDate = lc => julianDayToDate(longCountToJulianDay(lc))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1075,"value":"// little-endian, because you slice subsets from the little end… (sorry!)\nlongCountBases = [20, 18, 20, 20, 20]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1143,"value":"basesToValues = bases => bases.map((d, i, arr) => product(arr.slice(0, i)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1085,"value":"product = arr => arr.reduce((acc, cur) => acc * cur, 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":834,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":868,"value":"md`#### See also\n- [BabelStone Mayan Numerals](http://www.babelstone.co.uk/Fonts/Mayan.html), a font implementing the Unicode Mayan numeral block`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"todo = md`#### To-do\n- Animate\n- Check Cajori\n- Align vertically better\n- Animate addition / subtraction\n- Email Michael D. Coe??`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3a","stdlib":"1","titling":"auto"}} 32:"$31:body"