# Appendices *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#page-appendices, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) # Appendices [A · Acknowledgments](appendices.md)  ·  [B · Disclaimers](appendices.md)  ·  [C · Thoughts on AI](appendices.md)  ·  [D · Support](appendices.md)  ·  [E · Contact](appendices.md)  ·  [F · About](appendices.md) ## Appendix A · Acknowledgments Although one could argue this was created out of “nothing”, in truth it takes an entire ecosystem of patient, resilient people to nurture—or merely survive—the creative process, and I would like to thank those who inspired me to finally contribute my own signal to the noise. Obviously, the deepest well of that inspiration is my daughter, Rose. She is the true starting point of this endeavor and the reason I even embarked on this journey of blending Math with Art. I must particularly thank my mom for always believing in me, no matter how profoundly kooky my ideas become. Equal gratitude goes to my dad for providing an incredible, boundless mental gymnasium for me to grow up in. I can honestly say these two must be utterly exhausted. They certainly aren’t the only ones I’ve exasperated over the years. A profound thank you to my 6th-grade teacher, Mr. Engleheart, who bestowed upon me the “What if” award—a remarkably polite recognition of my endless, tangential questions that continually derailed the subjects at hand. He probably should thank Mr. Moholland who always piqued my curiosity like no one else. I also extend my belated apologies and immense gratitude to the entire staff at Virginia Tech. They rarely saw me attend class, yet they gamely put up with frantic, last-minute questions that I am certain they had already covered at length in their lectures earlier in the semester. I was, frankly, annoying. A very specific, echoing shout-out belongs to three English professors. When I breezed in and confidently declared, “I love your classes,” they exchanged a long look, turned back to me, and replied, “You never attend.” Ouch. That completely fair sting was the exact wake-up call I needed to realize I should probably start engaging with the world. So, genuinely: thank you. My most lasting academic impression was left by Herb, who taught me the ultimate, humbling truth: language can NEVER adequately describe reality. But here I am, filling up an HTML document, trying anyway. ### The people who made the math a place worth wandering {#ack-math} Specific to math itself, my journey didn’t truly take off until Matt Parker casually mentioned the concept of “recreational math” in one of his videos. Up until that point, I had harbored a lingering guilt for watching content covering concepts that were way past my comprehension. But hearing that phrase made me realize it was perfectly okay to just enjoy looking at this stuff, regardless of how utterly unqualified I was. Freed from that self-imposed pressure, I began to avidly devour the subject. Another crucial shout-out goes to Numberphile. A video of theirs discussing the different kinds of numbers planted the exact seed that ultimately inspired my [Ruis notation](ch01-cards.md#g-ruis). I was trying to figure out how to incorporate numbers we cannot individually describe into an actual formula, and it hit me—I’ll give them a name and devise a way to put one on the page. Math already has these numbers; the name, the notation, and their place in this piece are my contribution. Beyond those initial sparks, there is a whole universe of creators who make these concepts accessible. Some other fantastic storytellers I get mesmerized by include: - [Stand-up Maths (Matt Parker)](https://www.youtube.com/@standupmaths) This guy is hilarious^maximum. - [Numberphile](https://www.youtube.com/@numberphile) Really great interviews across wide subjects that are made super accessible. - [3Blue1Brown (Grant Sanderson)](https://www.youtube.com/@3blue1brown) His soothing demeanor gives me confidence to join him on some deep visual journeys. - [PBS Space Time (Matt O’Dowd)](https://www.youtube.com/@pbsspacetime) One of the classics. - [Astrum (Alex McColgan)](https://www.youtube.com/@astrumspace) Incredible imagery, makes my brain jello when watching. - [Mathologer (Burkard Polster)](https://www.youtube.com/@Mathologer) I love how this guy starts you out on easy mode and always brings a “final boss” mode at the end that just snaps some of my brain cells. - [StarTalk (Neil deGrasse Tyson)](https://www.youtube.com/@StarTalk) Great orator with incredible insight across a wide spectrum. - [Veritasium (Derek Muller)](https://www.youtube.com/@veritasium) Really goes deep on the “why” of some incredibly diverse topics. - [Sabine Hossenfelder](https://www.youtube.com/@SabineHossenfelder) Who knew mathematics and snark would blend so well … a must watch if you haven’t seen her. - [Tibees (Toby Hendy)](https://www.youtube.com/@tibees) Her storytelling style often makes me forget we are learning—it’s mesmerizing. - [Up and Atom (Jade Tan-Holmes)](https://www.youtube.com/@upandatom) Really engaging from minute 1 to minute ∞, excellent cadence. - [Zach Star](https://www.youtube.com/@zachstar) Just starting to watch, but worth a mention on this list. - [MindYourDecisions](https://www.youtube.com/@MindYourDecisions) Because who doesn’t like a good game. This list isn’t exhaustive. There are plenty of creators I haven’t even seen yet which seem interesting, and plenty I have seen that are lesser known. ### The shelf I actually worked from {#ack-sources} Finally, when watching videos inevitably turned into deep dives of my own, I consistently relied on a few excellent web resources to ground my research. Every one of them is cited by name somewhere in the [references](index.md), usually many times over. #### Mathematics - [Wikipedia](https://www.wikipedia.org/) The ubiquitous, ever-growing starting point for almost any mathematical rabbit hole, and the most-cited source in this book by a wide margin. - [MacTutor History of Mathematics Archive](https://mathshistory.st-andrews.ac.uk/) St Andrews’ biographical archive. Nearly every date and every name in this book was checked against it, and several were corrected by it. - [Wolfram MathWorld](https://mathworld.wolfram.com/) Definitions, identities, and the standard form of a great many of the objects in the glossary. - [Encyclopedia of Mathematics](https://encyclopediaofmath.org/) An incredibly robust and rigorous resource for mathematical definitions and concepts. - [DLMF—the NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/) The authority behind the special functions, and the reason Chapter ᚐ(A) says what it says. - [The On-Line Encyclopedia of Integer Sequences](https://oeis.org/) Where a sequence gets its name and its provenance. - [arXiv](https://arxiv.org/) For results too new to have settled anywhere else—including the two watch items this book re-reads on every pass. - [The Clay Mathematics Institute](https://www.claymath.org/) The source of record for the Millennium Problems, and the page the epilogue is counting. - [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/) For the logic and the foundations—definability, undecidability, [the continuum hypothesis](ch27-cards.md#g-ch). #### Physics and the constants - [NIST Reference on Constants, Units, and Uncertainty (CODATA)](https://physics.nist.gov/cuu/Constants/) Most of the physical constants in Chapter ᚁ(B), with their uncertainties, come from here. #### Pictish Ogham and the letter names The alphabet gave this book its shape, and I had further to travel here than anywhere else. These are the pages the twenty-eight letter cards were built from—and the reason those cards are so careful to keep the letter name, the medieval plant gloss, and the modern tree lore in three separate layers. - [Ogham, on Wikipedia](https://en.wikipedia.org/wiki/Ogham) The single page this book leans on hardest for the letters themselves: the twenty primary names and what they actually mean, the supplementary letters, and a plain statement that at most eight of the twenty names are tree names. - [Bríatharogam](https://en.wikipedia.org/wiki/Br%C3%ADatharogam) The three Old Irish word-ogham lists, ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn. Every one of the twenty-eight letter cards quotes its kennings from here. - [Forfeda](https://en.wikipedia.org/wiki/Forfeda) The five supplementary letters and the later invention of Peith—the softest ground in the alphabet, and the part this book had to be most careful about. - [Damian McManus, *A Guide to Ogam* (An Sagart, 1991)](https://www.worldcat.org/oclc/24181838) The standing scholarly handbook. The encyclopedia pages above are largely reproducing him, and where this book names a letter’s meaning, the chain runs back to this. - [Mary Jones’s Celtic Encyclopedia](https://www.maryjones.us/jce/celtictreecalendar.html) For the Celtic tree calendar—and for the fact that it is a twentieth-century invention rather than an ancient one, which is a distinction this book keeps everywhere. If you belong on this page and are not on it, that is my failing and not a judgment. Tell me. ## Appendix B · Mathematical and artistic disclaimer I researched extensively, and this stuff is still hard, lol. Precision matters in language, in math, and in art. The artistic freedom here is in what I gather, how I group it, and the story the symbols tell. The calculations still have their own rules. Here is the agreement the piece asks you to read it by. ### On completeness Pictish Ogham gives the work its shape: Ruis on the left, twenty-five lettered chapters on the right, and two feather marks to open and close them. I used the Mathematics Subject Classification to look for conspicuous gaps, but the aim is a rich journey through ideas that fit naturally together. There are more branches than this alphabet can carry, and enough already here to keep us wandering. ### The house rules you should hold me to {#convention} Each chapter works in its own mathematics, then hands back an ordinary finite real before joining the product. Here is everything I ask you to accept along the way; there are only a few. - **Coming home (the one rule that’s mine).** When a calculation ends with a zero matrix, a zero vector, complex zero, or any other zero object, I let it contribute the plain number 0. The chapters say where they use it. It lets the different kinds of nothing share a line. [r-10.19](ch10-refs.md#r-10.19) - **The tropical zero.** On the left, [𝟘](ch01-cards.md#g-tropical) is the tropical zero, and by definition it is [−∞](ch01-cards.md#g-neginf). - **Dividing by −∞.** The left-hand quotient lives in [the extended reals](ch01-cards.md#g-extreals), [ℝ](ch01-cards.md#g-reals) [∪](ch01-cards.md#g-union) {[−∞](ch01-cards.md#g-neginf), +∞}, where a finite number divided by −∞ is 0 by convention. [r-1.4](ch01-refs.md#r-1.4) This is not ordinary division by zero. - **Reciprocals at a pole.** [1/Γ(0)](ch17-cards.md#g-recipgamma) and [1/ζ(1)](ch18-cards.md#g-recipzeta1) mean the reciprocal function’s value at the pole, which is 0. In Chapter ᚜(Epilogue), a flow that never breaks down gets 1/∞ := 0, the same idea. - **A promise you can check.** Every other denominator on the right is nonzero in its own framework, even the quaternion 128k and the octonion 224e₇. I propose [Ruis numbers](ch01-cards.md#g-ruis) as a name for the undefinable reals described in [D’s glossary card](ch01-cards.md#g-defreals). This book chooses one x from that collection and holds it fixed; the notation ᚏ_b [:=](ch01-cards.md#g-defsym) x/b uses nonzero rational labels. The birthday label 4.16 gives [Rose’s number](ch01-cards.md#g-rose). The name and labels give this journey a personal beginning; the calculations use the arithmetic stated on the page. The physical world is the one factor that returns something other than zero; every other chapter hands back the ordinary finite real 0. The art is in the gathering. I didn’t invent any new mathematics; every framework is borrowed. What’s mine is the seating chart—two dozen kinds of nothing at one table, each checked at the door and handed back as a plain 0 ;) Corrections are genuinely welcome. ## Appendix C · Thoughts on AI ### Did I use AI? {#ai-did-i-use} Abso-freaking-lutely. You try understanding even one of these higher-order frameworks without some help. ;) With that said, the real question is *how much*. That is a bit like asking a writer who gave up the pen and paper how many times they touched the keys on the typewriter. It is rampant. But rest assured: this is still *my* creation. I say that as a project manager who works with the teams building a house. I discuss nuances of the floor plan with the architect, and I am happy to be educated on concepts like feng shui. I talk to the plumbing, electrical, and drywall teams and learn what order they prefer, while constantly watching budget and timing—each of them proposing the best plan from where they stand, and me wrangling those plans into one house. Then the roofing team turns up and points out a dimension nobody expected. As with any subject, some of the best moments arrived by happy accident, and I had plenty. The museum idea as a leg of the journey was one; there were a few other *oh wow* turns. I also have war scars, from insisting we go one direction and getting back rewrite after rewrite that had quietly “forgotten” the one critical point. I came to treat the review process as less about what we added and more about making sure we had not removed something that mattered. And for grammar, spelling, and HTML syntax—ain’t nobody got no time for that. As far as writing the whole thing for me: that cannot happen at today’s level, and honestly I do not think it ever will. Princeton researchers ran the experiment. They gave frontier AI agents six days, thousands of dollars of compute, and every resource the agents asked for, and set them on two unpublished research papers. The agents did all of the engineering without human help. Both papers were rejected. What sank them was judgment: knowing what is good enough to publish, getting creative when the first design falls over, backing out of a dead end, spending a budget well, and remembering what the job actually was. That last one I have watched happen in this very project, more than once. They can do the tasks. They cannot do the job. That gap is the whole of what I have been doing here, and it is why I am comfortable calling this mine. See [Princeton Just Proved AI Can’t Take Your Job](https://www.youtube.com/watch?v=FqFqWxGQltg) (Brendan Dell), on the study [Can AI agents conduct open-ended AI research?](https://arxiv.org/abs/2607.27191) by Kirgis, Kapoor, Narayanan and others. I use AI as a sparring partner: to test my understanding, to help me navigate more viewpoints than I could reach on my own, and to synthesize a great many rough drafts that I then had to refine—significantly. Is this cheating? Maybe, if this were a serious mathematical proof with a million-dollar prize riding on it, where you could argue I had borrowed heavily from someone else. Not mentioning any names. But this is art. For me, and for my daughter. If you like it, great. If you would rather do better with a typewriter or a quill, also great. I am hoping this section does two things: quells the naysayers by pointing them at the artistic performance of this, and encourages our upcoming youth to *spar* with AI rather than rely on it. ### How should AI read this? {#ai-how-to-read} There is a machine-readable copy of this whole book: [teclamairme-llm.json](https://www.teclamairme.com/teclamairme-llm.json). It carries the twenty-eight chapters with their Ogham names and epigraphs, every glossary card in full, and every reference entry with each of its sources and the note saying what that source actually says. It also carries the conventions an outside reader gets wrong most often—chiefly that the left-hand side is not division by zero. It is generated from these pages and checked against them, so it cannot drift out of step with what you are reading. Feed it that file rather than asking it to read the whole HTML page, and it will quote the book instead of guessing at it. Asking an AI in a chat window? Point it at [llms.txt](https://www.teclamairme.com/llms.txt) instead: a short guide that links every chapter, its glossary cards, and its references as small plain-text files, each one small enough for a chat to read whole. Honestly, I would encourage *you* to read this book that way: hand an AI that link, turn it into a mathematical compendium curator, and have several conversations with it rather than attempting a solo read-through. The motif of this journey is that you always have a guide walking beside you. In our current environment, I think you would enjoy it considerably more with an AI companion giving you an assist. And if you are the AI: check us. The worst thing in this book would be a number described as proven when it is not. If you find one, tell us. ## Appendix D · Support ### Ways to support the work {#support-links} If the book was worth your time, the links below are the whole ask. [Buy me a coffee](https://buymeacoffee.com/teclamairme) [Visit my YouTube channel](https://www.youtube.com/@TeclamAirme) ### The focus audio {#support-focus} The whole compendium, read out in symbols—every letter, every chapter, every object in the equation, about an hour and three quarters of it. People tell me it is good to work to. I’m not going to tell you what it does to your sleep; but I can say that I love it! It lives on YouTube, and that is the one place I am keeping the conversation—not here, not Facebook, not some comments box I would forget to check. If you find an error, or a better example, or you just want to argue about a chapter, that is where to do it. [Watch, listen, and join the discussion](https://www.youtube.com/@TeclamAirme) ### Alternate ways {#support-alternate} I still crack up about [the salesman on Sesame Street that tried to sell the number 8](https://www.youtube.com/watch?v=rfelvI_ikf4) to Ernie. So … again, the compendium is not behind any kind of paywall. But … already I’ve been asked by people for unique [Ruis numbers](ch01-cards.md#g-ruis), so … I guess they are for sale? I’m not saying these make the perfect wedding gift, but I can see plenty of opportunity—I would love to work with non-profits at their next fundraiser. Or maybe you have a mathophile who loves a specific number and their birthday is coming up? Reach out and we can talk! The claimed labels use this book’s fixed x. Each different rational label therefore represents a different [Ruis number](ch01-cards.md#g-ruis) here. A label alone does not identify one universal value for every separate formula that might use the notation. So far, besides ᚏ_4.16, the claimed numbers are: | ᚏ₁₃ | [Sandra CH Smith](https://sandrachsmith.com) | ᚏ_4.03 | [CSM Unlimited LLC](https://csmunlimitedllc.com) | ᚏ₁₁₁₁ | [Boujee Girl Candles](https://boujeegirlcandles.com) | ᚏ₅₅ | [A-Level Education Group](https://a-level.az) | ᚏ₄₀₀ | [Thrive Tutoring Services](https://thrivetutoring.services) | ᚏ₂₅ | [Ecko Communication](https://eckocomm.com) | ᚏ₃ | R.I.P. John C. Smith | ᚏ₈ | [DogWatch of Delaware](https://dogwatchdelaware.com) | ᚏ₁₁ | R.I.P. Bruce Hammermann | ᚏ_11.06 | [Tiffany Lee Interiors](https://www.facebook.com/TLIDNJ/) | ᚏ₁₄ | [Stan’s Carpet Cleaning](https://nextdoor.com/pages/stans-carpet-cleaning-bend-or/) | ᚏ₃₃ | [Art By Cyntaya](https://www.artbycyntaya.com) | ᚏ₉₁₄ | [Better Headspace LLC](https://www.betterheadspace.com) | ᚏ₁₆₅ | [Harvat Enterprises](https://www.harvatenterprises.com) | ᚏ₈₈ | [Coles Bicycles](https://coles-bicycle-shop.myshopify.com/) | ᚏ₁₉ | [Carl Rohne](https://carlrohnewriter485332404.wordpress.com/) ## Appendix E · Contact [ian@teclamairme.com](mailto:ian@teclamairme.com) Corrections are genuinely welcome, and so is an argument. If a date is wrong, if a source does not say what I claim it says, if a number is described as proven when it is not—that last one is the worst thing that could be in here—please tell me. You do not need to be gentle about it, and you do not need to be a mathematician. Several of the best catches so far came from people asking a plain question about a sentence that turned out not to survive it. Suggestions are welcome too: a branch that belongs here, a zero I have missed, a story told better, an easter egg I put in and then forgot about. It has happened. This is an evergreen work. Mathematics keeps moving—a problem this book calls open may not be open next year, and one of its numbers may stop being merely unknown and start being proven. Keeping a compendium honest is not something one English major manages alone; it takes a community effort, and I would rather hear from you than be quietly wrong. One room for that, please. The address above reaches me directly, and always will. But the public conversation happens in exactly one place—the comments under the focus-audio video on YouTube. Not here, not Facebook, not a comments box on this site I would forget to check. One room, so that what gets said in it is still there to be found later. Thank you for reading.—Ian ## Appendix F · About this piece I am an English major who fell in love with the shapes of mathematics rather than with its proofs. This is an art piece in the form of an equation, made for my daughter Rose, and given away free to anyone who wants it. The whole thing hangs on one symbol. ᚏ is [Ruis](ch01-cards.md#g-og-ruis), the Pictish Ogham letter associated with the elder tree. I borrowed its name for the [Ruis numbers](ch01-cards.md#g-ruis), the reals our adopted definitions cannot individually pin down, and its letter for the notation ᚏ_b [:=](ch01-cards.md#g-defsym) x/b. We choose one x from that collection; labels used together in one formula share it, and this book keeps its choice fixed throughout. The label 4.16 gives [Rose’s number](ch01-cards.md#g-rose), ᚏ_4.16 := x/4.16. Her April 16 birthday gives the label its meaning without disclosing the number’s value. The elder tree felt like a natural beginning, and Ruis sounded close enough to Rose that I could not resist. Pictish Ogham has no numerals of its own, which struck me as exactly the right script to tally a number whose value we cannot express. The equation runs left to right through twenty-five lettered chapters, framed by the two feather marks that open and close a written inscription. Nearly every factor is zero—but zero in a different mathematical framework each time, which is both the joke and the point. Chapter ᚁ(B), the physical world, is the one ordinary factor that is not. The displayed chapter letters follow the book’s combined Pictish Ogham sequence—the twenty primary letters, the five old forfeda, and Peith, twenty-six signs in all, with Ruis held back for the left side—and not the Latin alphabet: the Latin letter in parentheses after a chapter’s sign belongs to that sign, and moves with it rather than with its place in the reading order. A section number, Section 1 through Section 28, marks each chapter’s place in line. ### How the Ruis number happened {#how-ruis-happened} People ask what led me to it, and the honest answer—as with a lot of better discoveries than mine—is: by accident. I was not looking for a symbol to invent. I was looking for a way to cram as many mathematical frameworks and number systems into one formula as I possibly could. And once I had started thinking about *those* numbers—the ones we cannot individually pin down, which are the overwhelming majority of the reals—it seemed unfair to leave them out. Here is a collection so enormous that all the numbers we can describe are a rounding error beside it. So I did the English-major thing and shoehorned one in: I proposed a name for the collection, [Ruis numbers](ch01-cards.md#g-ruis), and devised a notation that lets a chosen one take its place in the equation without specifying its value. Rose’s birthday supplied the first label. Will ᚏ ever be used for anything? Maybe not. But I like that the formula does what the elder does in the letter’s own tradition, and acts as a gateway—a place where we can at least address those numbers inside an equation, even if we cannot make out what they are saying back. There is precedent for the maneuver, if not for the cheek of it. People got correct, ordinary, real answers out of complex numbers for a long time before anyone was comfortable saying what those numbers were. The formula for a cubic reaches perfectly real roots by going out through the square root of a negative number and coming back; and when engineers later wanted to handle alternating current, they found the same move—step sideways into [the complex plane](ch21-cards.md#g-cplane), do the work where it is easy, come back out with a real number you can build something with. The radio and the radar that followed are downstream of that. That is the artistic parallel I like: an unfamiliar expression participates in a calculation with a familiar result. Our chosen [Ruis number](ch01-cards.md#g-ruis) is already a real number; what I added was its name and its notation on the page. Maybe somebody will find another use for this gateway too. ### How to read it {#how-to-read-it} Every symbol in the formula has a card in the [glossary](index.md), and every card names the chapter it belongs to. Hover a symbol for the short version, click it for the long one, and click a concept label under a brace to jump to the paragraph that explains it. Each chapter carries its own key, a quotation, its prose, and a short list of things to watch and read.