Theory and Applications of Continued Fractions MATH X 451.50 | Fall 2022

For the Fall 2022 offering Dr. Michael Miller is offering a mathematics course on Theory and Applications of Continued Fractions at UCLA on Tuesday nights through December 6th. We started the first class last night, but there have been issues with the course listing on UCLA Extension, so I thought I’d post here for any who may have missed it. (If you have issues registering, which some have, call the Extension office to register via phone.)

For almost 300 years, continued fractions—that is, numbers representable as the sum of an integer and a fraction whose denominator is itself such a sum—have fascinated mathematicians with both their remarkable properties and their myriad applications in such fields as number theory, differential equations, and computer algorithms. They have been applied to piano tuning, baseball batting averages, rational tangles, paper folding, and plant growth … the list goes on. This course is a rigorous introduction to the theory and mathematical applications of continued fractions. Topics to be discussed include quadratic irrationals, approximation of real numbers, Liouville’s Theorem, linear recurrence relations and Pell’s equation, Hurwitz’ Theorem, measure theory, and Ramanujan identities.

Mike is recommending the Continued Fractions text by Aleksandr Yakovlevich Khinchin. I found a downloadable digital copy of the 1964 edition (which should be ostensibly the same as the current Dover edition and all the other English editions) at the Internet Archive at  Based on my notes, it looks like he’s following the Khinchin presentation fairly closely so far.

If you’re interested, do join us on Tuesday nights this fall. (We’ve already discovered that going 11 for 37 is the smallest number of at bats that will produce a 0.297 batting average.) 

If you’re considering it and are completely new, I’ve previously written up some pointers on how Dr. Miller’s classes proceed: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

Brodart Library Supplies for the Analog Zettelkasten Enthusiast

So you’ve taken the plunge and purchased an old school library card catalog, or maybe you want to but haven’t hit critical mass of cards to justify the purchase yet? Certainly you’ve found the traditional index card supplies still available at every office supply store on the planet, but did you know there’s still at least one company that supports libraries with custom card catalog supplies that you could use with your zettelkasten?

Brodart is a library services company based in Pennsylvania that supplies materials to institutional libraries that still has a variety of supplies not only for libraries and book lovers alike, but for amateur and professional zettelmacher(in) as well.

Most of their focus is on 3-by-5 inch index card sized material, but maybe with the re-popularization, they might add more support for the 4-by-6 inch card enthusiasts?

Perhaps if the demand for these older systems goes up, they’ll not only have more offerings, but the price will come down through economies of scale?

Let’s look at what they’ve got available.

Cards and Card Guides

On the card side, they’ve got a variety of options that aren’t as readily available at most office supply stores. If you’ve got an old school library card catalog with rods, you’re probably going to want cards with holes pre-punched. Of course they’ve got them in colors as well as without holes too.

With a sizeable card collection you’re likely to want some card guides, so they offer the traditional A-Z 1/5 Cut Card Guides as well as Blank Catalog Card Guides, with those holes pre-punched for convenience.

3x5" 1/3 cut manilla card guides with pre cut holes for separating your card sections

Dewey Decimal Catalog Card Guides

Most may already have an indexing system built into their system, but if you don’t and want to go with a classic Dewey Decimal set up, they’ve got you covered.

Dewey Decimal system manilla card with a tab that reads "000 General Works". The card has the BroDart logo and the number 24-111-101.

Perhaps you’ve got a sizeable digital card collection already, and have been jonesing to make the jump to analog? They’ve got printable card sheets so you can print out your digital cards relatively easily and continue without losing all that work. Or maybe you’re the mid century/ Umberto Eco purist who wants typewritten cards, but don’t want to retype them all? They’ve got both 4-up and 3-up versions as well.

A perforated sheet of paper with outlines for 4 3x5" cards with pre-drilled holes in each.
4-Up Catalog Card Sheets for Laser Printers

Let’s say you’ve got a long standing practice of making bibliographic cards. You need some cards to hold not only your meta data about the materials you’re reading, but you want to add your fleeting notes to them the way Luhmann and others have. Brodart has a wide variety of pre-printed cards that could serve this purpose. Some have printed sections which say “Date Loaned” and “Borrowers’s Name”, with sections for data below, but these could just as easily stand for page number and lined space for your important notes.

Brodart White Book Cards with Author, Title, Date Loaned, and Borrower’s Name

A 5x3" card that would appear in a library book with an empty section at the top followed by fields labeled "Author" and "Title". Below these are a two colum set of lined spaces under headings for "Date Loaned" and "Borrower's Name".

There are also a number of other versions of this sort of card depending on what you want. Try these or search for the many others which may suit your fancy:

Slip Boxes

Maybe you haven’t made that slip box purchase yet, but want something shiny and new? Brodart has you covered here as well. They’ve got a few different options for a small desktop slip box or a fully modular system that you can add to over time.

Stand alone boxes

Brodart has at least two desktop boxes, with 12 and 9 drawers respectively.

A wooden table top library card catalog with drawers in a 4x3 configuration. Each drawer has a metal pull with a label slot and at the bottom a removable card file rod.

Modular Boxes

Want to design your own system that’s expandable with your card collection? They’ve got a five drawer wide system with options for 1, 2, or 3 row tall sections that you can build up to suit your needs. Start with their table and legs, add a one or more sections of card files, and then top it off with a cover. If you’d like, they’ve also got an interstitial piece with drawer pulls so that you’ve got a writing surface built into your zettelkasten. Build that system up to your ceiling!

A modular 3x5 drawer card catalog box. The top is open so as to accomodate other similar modular boxes or a woodenn cap top.

4-by-6 inch Card Boxes

Brodart is a bit thin on the 4-by-6 inch category, but for the beginning zettelmacher(in), they do have some nice sized, portable, archive quality boxes you might like to start your collection. See their Postcard Boxes.

Other Options

Of course there are lots of other options in the space. Some of these box systems can become pretty expensive, and for the price you might be as well off purchasing a used card catalog which you can restore  or you can find restored ones online. Some of them even go to the level of fine furniture and can quickly go for over $5,000.00.

If you prefer the vintage 20 gauge steel esthetic (you know I do!), you can find lots of used, but still great condition slip boxes online in places like eBay or on Craigslist.

I and others have written some advice about other card storage options on a Reddit community targeted at analog zettelkasten in the past.

What do you use? What do you want to use? Are you going to custom build your own? Have you seen other companies like Brodart that still support the manufacturing of these sorts of tools for thought? Please share your ideas and supplies below.

A Small 10,000+ Annotations Party

I recently hit the 10,000+ annotations mark on the fantastic Hypothes.is platform, and in celebration, the kind team at Hypothesis sent me a care package with a lovely card, some sticky flags, some stickers, and some chocolate to see me through the next 10,000.

If I’m honest, I get so much value and joy out of annotating online, I should have been the one to send them the care package.

Thanks Hypothes.is! I’ll see you in the margins.

Looking for books with wider margins for annotations and notes

With the school year starting and a new slew of books to be purchased and read, I’ve been looking for books, particularly popular “classics” or “great books” that are published either with larger margins or even interleaved copies (books in which every other page is blank and meant for writing extensive notes).

Within the bible publishing space (and especially for study bibles), these features seem a lot more common as people want to write more significant marginalia or even full pages of text against or opposite of what they’re reading within the book itself. Given the encouragement many teachers give for students to actively annotate their books for class discussions as well as people commonly doing this, why isn’t it more common for them to recommend or require texts with ample margins?

Almost all of the published mass market paperbacks I see of series like Penguin Classics or Signet classics have the smallest possible margins and no interlinear space for writing notes directly in books. Often hard covers will have slightly larger margins, but generally most publishers are putting 1/2 inch or 3/4″ margins on their classics series (Penguin Classics, Everyman Library, Signet (a paltry 1/4 inch usually), Library of America, Norton Classics, Wordsworth Classics, Dover, etc.)

For those wanting ample margins for active “reading with a pen in hand” are there any publishers that do a great job of wider margins on classics? Which publishers or editions do others like or recommend for this sort of reading?

Are there any book sales platforms that actually list the size of margins of their books? (I never seen one myself.)

What can consumers do to encourage publishers to change these practices?

I’ve seen only a few select titles from very few publishers that do things like this. Examples include:

  • Gladius Books’ Huckleberry Finn
  • Annotate Books’ Marcus Aurelius, which has 1.8″ lined margins
  • The Folio Society has slightly better margins on their texts, but they’re generally larger hardcover collectors’ editions that are dramatically more expensive than is practicable for students on a budget. ($80+ versus $5-10)

If I can’t find anything useful, I’m tempted to self-publish custom versions of wide margin or interleaved books otherwise. Something in the inch to inch and a half margin size for commonly used texts in literature classes should be much more commonplace.

Usually once a tag on my website has more than a couple hundred entries, I convert it into a category. This one was long overdue. This morning I’ve converted the “note taking” tag into a category and moved a bunch of material on commonplace book and zettelkasten traditions over to it. 

If you’ve been following this idea here, it’s time to update your feed.

How to Make Notes and Write, a handbook by Dan Allosso and S.F. Allosso

A new handbook on note making and writing

I wasn’t expecting it until next week or shortly thereafter, but just in time for the new academic year, Dan Allosso has finished a major rewrite on his and S.F. Allosso’s earlier edition of A Short Handbook for writing essays in the Humanities and Social Sciences. This expanded edition has several new chapters on note making (notice that this is dramatically different than note taking) using a zettelkasten-based (or card index or fichier boîte if you prefer) approach similar to that practiced by Beatrice Webb, Marcel Mauss, Claude Lévi-Strauss, Roland Barthes, Hans Blumenberg, Mortimer J. Adler, and Walter Benjamin among many others.

The focus of the book is on note making for actively producing tangible outputs (essays, papers, theses, monographs, books, etc.), something on which a few recent texts in a the related productivity space haven’t delivered. While ostensibly focused on the humanities and social sciences in terms of examples, the methods broadly apply to all fields. In fact, some of the methods draw historically on some of the practices fruitfully used by Bacon, Newton, Leibnitz, Linnaeus, and many others in the sciences since.

This isn’t your father’s note making system…

While many students (especially undergraduates and graduate students) may eschew this sort of handbook as something they think they “already know”, I can assure you that they do not and will benefit from the advice contained therein, particularly the first half. I’ve often heartily recommended Sönke Ahrens’ book How to Take Smart Notes: One Simple Technique to Boost Writing, Learning and Thinking to many in the past, but I think Allosso’s version, while similar in many respects, is clearer, shorter, and likely more easily realized by new practitioners.

There’s more detail in Dr. Allosso’s announcement video:

Availability

How to Make Notes and Write is available at Minnesota State’s Pressbooks site for reading online, or download as a .pdf or .epub. If you’d like a physical copy, they’re also available for purchase on Amazon.

For those in the educational spaces, Dr. Allosso has given the book a Creative Commons license (CC BY-NC-SA 4.0), so that people can use it as an Open Educational Resource (OER) in their classes and work.

For teachers who are using social annotation with tools like Hypothes.is in their classrooms, Allosso’s book is an excellent resource for what students can actively do with all those annotations once they’ve made them. (Here’s a link to my annotated copy of a recent working draft if you care to “play along”.)

† Unless of course your father happens to be Salvatore Allosso, but even then…

Hypothes.is, a web annotation tool, as an off-label zettelkasten?!

It wasn’t built specifically to be used as one, but is anyone actively using Hypothes.is as an off-label zettelkasten just by itself?

The platform has most of the basic functionalities one would want for a digital ZK:

  1. Simple note taking,
  2. Notes are editable and update-able,
  3. There’s a tag functionality for indexing notes,
  4. One can add links to other notes to cross link them if they liked,
  5. There’s a reasonably good search functionality,
  6. Data export is available if you want to move

The interesting piece is that if many are doing this in public, then folks can reply to others’ notes and even cross-link their public notes. (They’ve got the ability to have public and private notes, as well as groups for collaboration, which are functionalities most ZK don’t have as well.)

I was reminded of this potential off-label use case when someone replied to an older note with a quote/comment and it nudged me to add a note to link the two together. Who’s up for a public, social zettelkasten practice?

For those who need an example to look at, try my “digital ZK”: https://hypothes.is/users/chrisaldrich or their public timeline: https://hypothes.is/search.

I know some have mentioned Hypothes.is as an annotation tool for their note taking before. There are a few who use it as an online platform for notes and then they leverage the platform’s API or feeds to export their data to their tool of choice. (I’ve used Hypothes.idian for Obsidian to do just this.)

I’m curious if anyone has tried building a digital public zettelkasten on WordPress in general or using using the Slippy plugin in particular?

I’m thinking it may be an interesting experiment, particularly using it in combination with the Webmention plugin to get replies/responses for crosslinking with others’ ideas on the web. This could allow one not only to communicate with other their own slip box, but slip boxes to communicate with each other.

Call for Model Examples of Zettelkasten Output Processes

Perhaps too much of the resurgence of the zettelkasten idea and the online space about it is focused on what a zettelkasten is or how it should be done. After this, descriptions of the process of collecting material for one’s zettelkasten is followed by using it to generate new ideas and thought, though this last part is relatively sparce in comparison. Very little of the discussion or examples I’ve seen in online fora, social media, websites, and the blogosphere is focused on actively using them for creating actual long form output.

As Luhmann’s (all-too-frequently used) example is so powerful, I think it would be massively helpful if users had stronger examples of what these explicit creation workflows looked like, particularly at the longer end of creation of chapters or even book length spaces. Are there any detailed posts, videos, other media about how one approached this problem? What worked well? What didn’t? What would you do differently next time? Have you done this multiple times and now settled into something you feel is most efficient? Is your process manual/digital? What tools helped along the way for laying out and doing the actual stitching together and editing? Would you use them again or try something else? Have you experimented with different methods or practices?

Here, I’m looking for direct and actual experience; I’m explicitly not looking for “this is how I would do it” responses.

Because it’s much easier and far more successful for humans to imitate the practice of others than to innovate it for themselves, I’m ultimately looking not for outlines of what people recommend, but public examples of the practices in progress. Who can show their actual “receipts” and in a reasonably linear and practical way for others to follow? We suffer from a lack of these practices being visible online as most aren’t. Further even the digital ones aren’t public, or if they are, they aren’t well known.

As an example of the broader problem, I’ve yet to see a week go by that someone doesn’t read Ahrens’ generally excellent book, but in posting online they still seem lost in attempting to put the lowest level ideas into active practice.

Personally, I use Hypothes.is as a digital tool for the majority of my note taking. One could follow my feed and see it in real time if they choose. There are benefits for this public practice and I’m aware that many people follow this feed of notes out of curiosity. I’ve even gotten emails from folks indicating that they’ve learned some interesting things for use in their own practices. Sadly, the follow up of revision, cross linking, active indexing, and subsequent growth isn’t public (yet), though the platform I’m using is open to active public conversation and commentary, which is a useful side benefit. I have seen a few other public examples of others’ practices, some in video form, though this can be dull as the time and effort is work and doesn’t make for powerful entertainment because it isn’t. Still these public examples can be far more powerful than some of the explanations I’ve seen, especially for beginners who don’t comprehend the long term benefits (surprise, serendipity, insight, emergent creativity) and who generally focus on the minutiae for lack of direct experience.

On the creation portion, I’m currently experimenting with carrying out the original  instructions of Konrad Gessner, an early zettelmacher, as laid out in his book Pandectarum sive Partitionum Universalium. (1548. Zurich: Christoph Froschauer) and hope to report back shortly about the experience. 

Call for explicit examples

So where are the examples? Show us your receipts. Who’s doing this in public that people can follow? Who can be imitated until people have the experience(s) to do this more easily on their own? Let’s collect some of the best (or at least extant) examples for sharing with others. 

Once we’ve got some concrete examples then we can study them and iterate on them.

Too many people seem intent on potentially wasting their time by innovating on a practice they haven’t even tried because someone in the productivity space (who usually hasn’t tried it either) wrote a page long post saying it would be a good idea. I know we can do far better.

Independently-hosted web publishing

Bookmarked Independently-hosted web publishing by Daniel Villar-Onrubia & Victoria I. MarínDaniel Villar-Onrubia & Victoria I. Marín (Internet Policy Review Volume 11, Issue 2 DOI: 10.14763/2022.2.1665)
The term independently-hosted is used here to describe online publishing practices that utilise the World Wide Web (hereafter the Web) as a decentralised socio-technical system, where individuals and communities operate as the owners or controllers of the online infrastructures they use in order to share content. Such practices may be adopted as an alternative of, or as a complement to, the use of centralised content-sharing systems that belong to and are entirely operated by third parties. The term “publishing” is used here in a rather inclusive way and refers to the act of making content available online, rather than being restricted to the editorial processes that characterise, for instance, academic publishing.
DOI: 10.14763/2022.2.1665
Great to see IndieWeb.org cited in the academic literature.
Is anyone in the or space using @tinysubversionsHometown fork of to create small “local only” posting spaces for their classes? Are there any inexpensive hosts that have one click installs/setups for this? Screen capture of paragraph that reads: "In August 2018, Kazemi created his own Mastodon server (an “instance”) called Friend Camp. But he didn’t want it to be a popular instance — he wanted to run a small social network, with under 100 users. The goal was to foster community-related discussion and attain a sense of “group cohesion.” The following year, based on his experience of running Friend Camp, Kazemi forked Mastodon into a new software package he called Hometown. One of its main features is “local only posting,” which gives users the option of not federating their posts." The last line is highlighted in yellow.
Replied to a tweet by Codex Editor (Twitter)
To break your literacy boundaries, try taking a look at Lynne Kelly‘s work on orality and memory (Knowledge and Power, Songlines, et al.). Duane Hamacher, et al. have a great new book out as well. And try Ong’s work on orality too. There are lots of non-literate tools for thought hiding out there.
Replied to Note-taking: A Research Roundup by Jennifer GonzalezJennifer Gonzalez (Cult of Pedagogy)
A summary of 8 best practices in note-taking, straight from the research.
It’s been a few years since this was originally posted and there have been some interesting developments in note taking apps and software which have been shifting some of the focus in the space. The primary change is the popularization of the German idea of the Zettelkasten which grew out of the commonplace book tradition from antiquity and the early Renaissance.

The most detailed form of the idea can be found in Sönke Ahrens’ book How to Take Smart Notes, which also looks closely at much of the note taking and psychology related research over the past several decades. While he frames the method in terms of writing and creation as the end goal, much of the method dovetails with Bloom’s Taxonomy as I’ve outlined. It could also be framed as Cornell Notes with a greater focus on atomic notes that are highly linked and thereby integrating a student’s new knowledge with their prior knowledge.

I’d love to see more educators scaffolding the use of this note taking tool in their classes, especially in high school and undergraduate education.

Cross reference: https://boffosocko.com/tag/note-taking/

The Zettelkasten Method of Note Taking Mirrors Most of the Levels of Bloom’s Taxonomy

Between zettels (the space where new ideas often occur) this morning I realized that there is a high correlation to Bloom’s Taxonomy and the refined methods of keeping a zettelkasten, particularly as delineated by Dr. Sönke Ahrens in his book How to Take Smart Notes. As a result, even if one’s primary goal isn’t writing or creating as framed by Ahrens’, the use of the system as a pedagogical tool can be highly effective for a variety of learning environments.

Recall briefly that Bloom’s Taxonomy levels can be summarized as: remember, understand, apply, analyze, evaluate, and create.

Outlining the similarities

One needs to be able to generally understand an idea(s) to be able to write it down clearly in one’s own words. This is parallel to creating literature notes as one reads. Gaps in one’s understanding will be readily apparent when one realizes that they’re not able to explain an idea simply and clearly.

Regular work within a zettelkasten helps to reinforce memory of ideas for understanding, long term retention, and the ability to easily remember them. Many forms of zettelkasten software have functionality for direct spaced repetition if not dovetails for spaced repetition software like Anki or Mnemosyne.

Applying the knowledge to other situations happens almost naturally with the combinatorial creativity that occurs within a zettelkasten. Raymundus Llullus would be highly jealous here I think.

Analysis is heavily encouraged as one takes new information and actively links it to prior knowledge and ideas; this is also concurrent with the application of knowledge.

Being able to compare and contrast two ideas on separate cards is also part of the analysis portions of Bloom’s taxonomy which also leads into the evaluation phase. Is one idea better than another? How do they dovetail? How might they create new knowledge? Juxtaposed ideas cry out for evaluation.

Finally, as argued by Ahrens, one of the most important reasons for keeping a zettelkasten is to use it to generate or create new ideas and thoughts and then use the zettelkasten as a tool to synthesize them in articles, books, or other media in a clear and justified manner. 

Other examples?

I’m curious to hear if any educators have used the zettelkasten framing specifically for scaffolding the learning process for their students? There are some seeds of this in the social annotation space with tools like Diigo and Hypothes.is, but has anyone specifically brought the framing into their classes?

I’ve seen a few examples of people thinking in this direction and even @CalHistorian specifically framing things this way, but I’m curious to hear about other actual experiences in the field. 

The history of the recommendation and use of commonplace books in education is long and rich (Erasmus, Melanchthon, Agricola, et al.), until it began disappearing in the early 20th century. I’ve seen a few modern teachers suggesting commonplaces, but have yet to run across others suggesting zettelkasten until Ahrens’ book, which isn’t yet widespread, at least in the English speaking world. And even in Ahrens’ case, his framing is geared specifically to writing more so than general learning and education.

Featured image courtesy of Vanderbilt University Center for Teaching (CC BY 2.0

Contextualizing Cornell Notes in the Note Taking Traditions

Have we lost too much of the contextual value of what Cornell notes were originally designed for by Walter Pauk in the 1950’s? Or are we not taking the idea far enough into the writing realm?

Cornell notes come from a time closer to the traditional space of commonplace books, academic thinking, and note taking that was more prevalent in the early 1900’s and from which also sprang the zettelkasten tradition. I can’t help but be reminded that the 10th edition of Pauk’s book How to Study in College (Wadsworth, 2011, p.394), which helped to popularize the idea of Cornell notes with the first edition in 1962, literally ends the book with the relationship of the word ‘topic’ by way of Greek to the Latin ‘loci communes‘ (commonplaces), though it’s worth bearing in mind that it contains no discussion of the commonplace book or its long tradition in our intellectual history.

One was meant to use Cornell notes to capture broad basic ideas and facts (fleeting notes) and things to follow up on for additional research or work. Then they were meant to be revisited to focus on creating questions that might be used for spaced repetition, a research space that has seen tremendous growth and advancement since the simpler times in which the Cornell note taking method was designed.

Additionally one was meant to revisit their notes to draw out the most salient points and ideas. This is part of the practice of taking the original ideas and writing them out clearly in one’s own words to improve one’s understanding of the material. Within a zettelkasten framing, this secondary review is part of the process of creating future useful literature notes or permanent notes that one might also re-use in their future writing and thinking.

Missing from the Cornell notes practice but more directly centered in the zettelkasten practice is taking one’s notes and directly linking them to other related thoughts in one’s system. This places this method closer to the commonplace book tradition than the zettelkasten tradition.

While a more basic and naïve understanding of Cornell notes in current academic environments still works on many levels, students and active researchers might be better advised to look at their practices in view of broader framings like that of Sönke Ahrens’s book How to Take Smart Notes: One Simple Technique to Boost Writing, Learning and Thinking.

It also bears noting that one could view the first stage of Cornell notes in light of the practice of keeping a waste book and then later transferring their more permanent and better formed ideas into their commonplace book.

Similarly one might also view full sheets of finished Cornell notes as permanent notes mixed in amidst fleeting notes and held together on pages rather than individual cards. This practice sounds somewhat similar in structure to Sönke Ahrens’s use of Roam Research to compile multiple related ideas in individually linked blocks on a single page holding them together in a pseudo-project page for more immediate and potentially specific future use.