{"id":209,"date":"2026-07-10T12:19:10","date_gmt":"2026-07-10T12:19:10","guid":{"rendered":"https:\/\/tracefleemangarcia.com\/trXiv\/?p=209"},"modified":"2026-07-10T12:19:10","modified_gmt":"2026-07-10T12:19:10","slug":"geomancy-and-matrix-algebra","status":"publish","type":"post","link":"https:\/\/tracefleemangarcia.com\/trXiv\/2026\/07\/10\/geomancy-and-matrix-algebra\/","title":{"rendered":"Geomancy and matrix algebra"},"content":{"rendered":"\n

Mathematical approaches to geomancy usually focus on its binary aspects. This is probably just Maslow’s hammer<\/a>: our contermpoary lives are based around digital technology, and we see that technology in historical practices. Shoddy scholarship about historical transmission<\/a> aside, I think this is less than ideal for a couple of reasons. First, historical<\/em> geomancy was clearly understood in terms of modular arithmetic and parity. The same method for punctuation was used in sortes<\/em> works like the Prenostica Socratis Basilei<\/em>1<\/a><\/sup> to perform addition modulo 10.2<\/a><\/sup> Second, historical geomancers never bothered with arithmetic with figures that had less than or more than four rows.3<\/a><\/sup> So, historical geomancy didn’t function like binary arithmetic.<\/p>\n\n\n\n

Let me introduce a different mathematical approach, which I think is closer to the mark. First, instead of figures-as-numbers, let’s interpret them as arrays<\/em> of numbers. Caput draconis<\/em>, then, might be<\/p>\n\n\n\n

[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>\\begin{bmatrix} \n1 & 0 & 0 & 0\\\\ \n0 & -1 & 0 & 0\\\\ \n0 & 0 & -1 & 0\\\\\n0 & 0 & 0 & -1\n\\end{bmatrix}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

It might not be incredibly clear why I chose to represent passive\/active lines as -1 and 1, or why I chose to put them on the diagonal. However, there’s a good reason for this. Because of the way normal matrix multiplication<\/a> works, this is actually equivalent to geomantic conjugation. Here is Caput draconis<\/em> conjugated with Amissio<\/em>, giving Puer:<\/em><\/p>\n\n\n\n

[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>\u00d7<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\begin{bmatrix} \n1 & 0 & 0 & 0\\\\ \n0 & -1 & 0 & 0\\\\ \n0 & 0 & -1 & 0\\\\\n0 & 0 & 0 & -1\n\\end{bmatrix} \\times \\begin{bmatrix} \n-1 & 0 & 0 & 0\\\\ \n0 & 1 & 0 & 0\\\\ \n0 & 0 & -1 & 0\\\\\n0 & 0 & 0 & 1\n\\end{bmatrix} = \\begin{bmatrix} \n-1 & 0 & 0 & 0\\\\ \n0 & -1 & 0 & 0\\\\ \n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & -1\n\\end{bmatrix}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

There is already a nifty notation for a diagonal matrix<\/a>, which is a matrix that has zero entries everywhere except the diagonal. Here’s the same conjugation as above:<\/p>\n\n\n\n

diag<\/mtext>(<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>)<\/mo>\u00d7<\/mo>diag<\/mtext>(<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>1<\/mn>)<\/mo>=<\/mo>diag<\/mtext>(<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>,<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>)<\/mo><\/mrow>\\textup{diag}(1, -1, -1, -1) \\times \\textup{diag}(-1, 1, -1, 1) = \\textup{diag}(-1, -1, 1, -1)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

We can also use the Hadamard product<\/a> on column vectors and get something damn near equivalent to how conjugation is represented historically:<\/p>\n\n\n\n

[<\/mo>1<\/mn><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>\u2299<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>\u2212<\/mo>1<\/mn><\/mrow><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mrow>\\begin{bmatrix}\n1 \\\\\n-1\\\\\n-1\\\\\n-1\n\\end{bmatrix} \\odot \\begin{bmatrix}\n-1 \\\\\n1\\\\\n-1\\\\\n1\n\\end{bmatrix} = \\begin{bmatrix}\n-1 \\\\\n-1\\\\\n1\\\\\n-1\n\\end{bmatrix}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

The question is: why would we?<\/p>\n\n\n\n

The interpretation of historical mathematics<\/h2>\n\n\n\n

The simple fact of the matter is that historical geomancers did not think of geomancy as binary arithmetic. Neither did they think that it was a sort of matrix multiplication. In fact they had no knowledge of binary numerals, nor of matrices.<\/p>\n\n\n\n

Most mathematicians are naive little-p platonists<\/a>, that is, they believe that mathematical objects exist independently of us and outside of language. Or , at least, that’s how they talk: one uses verbs like ‘discover’ to describe mathematical achievements, not ‘invent’. And this is perfectly fine: I myself am a platonist, I suppose. However, this is not useful for historical research. If mathematical objects exist ‘out there’, they are timeless (Zalta uses the term ‘nonspatiotemporal<\/a>‘). Yet as historians we are concerned with temporal and spatial objects. <\/p>\n\n\n\n

Historians of mathematics are often too concerned with priority, even priority that had no historical influence.4<\/a><\/sup> Importantly, mathematical objects are fungible, if not metaphysically, then in practice: the same ‘0’ I use is the same ‘0’ you use. Historical objects are not. Two witnesses of the same work are not the same, but have distinct biographies<\/a>. Ideas are historical objects, too: their content matters, but their context matters too, maybe even more than the former.<\/p>\n\n\n\n

It might be interesting to understand the mathematical underpinnings of a given historical practice, but we must remain steadfast methodological antiplatonists<\/em>. For the purposes of historical research, pre-colonial Mayas, medieval Indians, and modern Westerners do not use the same ‘0’ but historically distinct ones with their own biographies, their own transmissions, their own lines of flight. <\/p>\n\n\n\n

Footnotes<\/h2>\n\n\n
  1. See Guardo, Albero Alonso. Prenostica Socratis Basilei: \u00c9tude, \u00e9dition critique et traduction<\/em>. Classiques Garnier, 2015. \u21a9\ufe0e<\/a><\/li>
  2. This is not strictly accurate — since this text predates the introduction of 0 to Europe, so it was held that x<\/mi> mod <\/mtext>x<\/mi>=<\/mo>x<\/mi><\/mrow>x \\textup{ mod }x = x<\/annotation><\/semantics><\/math>. The same goes for prognostic spheres, which do not show 0 but do show their modulus. \u21a9\ufe0e<\/a><\/li>
  3. This again is not strictly accurate; a Byzantine manuscript<\/a> (Harley MS 5596) seems to witness a simplified form of geomancy with three row figures. The 19th century Oraculum<\/em> attributed to Napleon for some reason<\/a> uses five row figures. My point still stands, though, as there doesn’t seem to be an arbitrary length of binary numbers used anywhere. \u21a9\ufe0e<\/a><\/li>
  4. For example, Ares, Juan, et al. “Who discovered the binary system and arithmetic? Did Leibniz plagiarize Caramuel?.” Science and Engineering Ethics<\/em> 24.1 (2018): 173-188. Conspiracy theories about plagiarism notwithstanding. \u21a9\ufe0e<\/a><\/li><\/ol>\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    Mathematical approaches to geomancy usually focus on its binary aspects. This is probably just Maslow’s hammer: our contermpoary lives are based around digital technology, and we see that technology in historical practices. Shoddy scholarship about historical transmission aside, I think this is less than ideal for a couple of reasons. First, historical geomancy was clearly…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":"[{\"content\":\"See Guardo, Albero Alonso. Prenostica Socratis Basilei: \u00c9tude, \u00e9dition critique et traduction<\/em>. Classiques Garnier, 2015.\",\"id\":\"92ecb05d-8861-4d79-b1d7-fd96b78a0236\"},{\"content\":\"This is not strictly accurate -- since this text predates the introduction of 0 to Europe, so it was held that x<\/mi> mod <\/mtext>x<\/mi>=<\/mo>x<\/mi><\/mrow>x \\\\textup{ mod }x = x<\/annotation><\/semantics><\/math>. The same goes for prognostic spheres, which do not show 0 but do show their modulus.\",\"id\":\"a060e4ba-02ee-4ff6-8ce0-4d93cb445f29\"},{\"content\":\"This again is not strictly accurate; a Byzantine manuscript<\/a> (Harley MS 5596) seems to witness a simplified form of geomancy with three row figures. The 19th century Oraculum<\/em> attributed to Napleon for some reason<\/a> uses five row figures. My point still stands, though, as there doesn't seem to be an arbitrary length of binary numbers used anywhere. \",\"id\":\"18b86124-26e0-4096-8402-acf49b53e9c6\"},{\"content\":\"For example, Ares, Juan, et al. \\\"Who discovered the binary system and arithmetic? Did Leibniz plagiarize Caramuel?.\\\" Science and Engineering Ethics<\/em> 24.1 (2018): 173-188. Conspiracy theories about plagiarism notwithstanding.\",\"id\":\"be66b3fc-305c-4d1c-bcd3-59790627f791\"}]"},"categories":[1],"tags":[],"class_list":["post-209","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/posts\/209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/comments?post=209"}],"version-history":[{"count":1,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/posts\/209\/revisions"}],"predecessor-version":[{"id":210,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/posts\/209\/revisions\/210"}],"wp:attachment":[{"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/media?parent=209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/categories?post=209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tracefleemangarcia.com\/trXiv\/wp-json\/wp\/v2\/tags?post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}