Tag: mathematics

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Ancient Web: DPGraph Drew Eight-Dimensional Math in Assembly Language

DPGraph is a mathematical graphing program from the late 1990s with a specification that sounds increasingly strange the longer you read it. It draws dynamic and interactive mathematical visualizations extending from ordinary two-dimensional plots through what its creator describes as eight-dimensional graphs, and the whole program was written in assembly language.

Visit the DPGraph website

Dave Parker released DPGraph in 1999, and the surviving update history still traces changes from that first release forward. The program was built for interactive mathematics and physics visualization: equations, animated surfaces, transformations, implicit plots, and other objects that are easier to understand when they can be manipulated instead of frozen into a textbook image.

Then there is the implementation detail. DPGraph’s site says the program was written entirely in assembly for speed. That was not unheard of in the 1990s, but an entire high-level visualization application written that way is still a pretty aggressive engineering decision. Modern software stacks often require several layers of frameworks before a window appears. DPGraph went in the opposite direction.

The site’s own usage claim is equally ambitious: it says more than two million mathematicians, physicists, teachers, and students at more than a thousand schools were licensed to use the software. That number is the site’s claim rather than an independently audited statistic, but it indicates the scale Parker believed the program had reached.

The old compatibility notes are now part of the attraction. DPGraph and its viewer were distributed for generations of Windows stretching from the Windows 95/NT era into Windows 10, with references to running under Linux and emulation as well. The viewer could also be used to present graphs through Web pages, a reminder of the period when downloadable helper applications and browser-linked viewers were normal parts of publishing interactive material online.

The latest prominent news on the site dates to 2019, and Parker now describes DPGraph as freeware with no foreseeable updates. That is not necessarily a sad ending. A piece of specialist software can be finished without becoming useless.

For software historians, the site preserves release notes, documentation, examples, downloads, and the author’s own description of what he built. For teachers and mathematically curious visitors, it remains a doorway into a visualization tool that came from a very different software culture.

It also illustrates one of the recurring patterns in CacheRat’s 1,967 Ancient Web Domains research list: small software sites can outlive entire commercial distribution ecosystems because the author kept the files and documentation in one stable place.

The dimensions may be difficult to picture. The old website explaining them is pleasantly straightforward.

Explore DPGraph, its history, documentation, and downloads

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Ancient Web: Blackjack in Color Turned Card-Counter Math Into Visual Charts

Blackjack in Color looks like somebody decided that tables of gambling probabilities were not nerdy enough and therefore needed thousands of simulations, distributions, and colored charts. The result is much more interesting as a mathematical Web artifact than the casino subject might suggest.

Visit Blackjack in Color

The site is built around visual explanations of probability. Instead of reducing everything to a slogan about whether a particular system “works,” it shows distributions, frequencies, count behavior, and the uncomfortable fact that random processes remain noisy even when the underlying mathematics is understood.

That makes the site useful as an example of simulation-based explanation. A chart can show something that a single expected-value number hides: how wide the range of outcomes is, how often unusual runs occur, and how long short-term results can wander away from an average.

The subject happens to be blackjack, but the presentation belongs to a broader tradition of independent technical websites. Somebody has a specialized question, writes software to investigate it, generates data, then publishes enough of the machinery that other interested people can inspect the reasoning instead of accepting a conclusion on faith.

That approach was especially valuable on the older Web because specialist information was often scattered between books, forums, and private correspondence. A purpose-built website could assemble the charts, terminology, and explanations into one stable location.

There is also a good design lesson in the name Blackjack in Color. The visualizations are not decorative dashboard furniture. Color is used because the site is trying to make relationships in a large numerical subject easier to see. The graphics support the explanation.

For a reader in 2026, the interesting part is not a promise of easy money. Randomness does not become obedient because a website drew a graph. The value is seeing a niche enthusiast treat probability seriously enough to model it, visualize it, and document what the results actually look like.

That is the old specialist Web at its best: a narrow subject, a technically minded author, and a pile of work that exists because somebody wanted the answer badly enough to calculate it.

Explore the simulations and charts at Blackjack in Color

Even if you never sit at a blackjack table, the site is a useful museum of applied probability with unusually colorful exhibits.

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Ancient Web: NYU Put Archimedes’ Cattle Problem on the Web

Archimedes managed to make counting cattle unpleasant enough that mathematicians were still talking about it more than two thousand years later. An old NYU mathematics page preserves the statement of the Cattle Problem in exactly the kind of university-hosted Web format that deserves to survive.

Read Archimedes’ Cattle Problem at NYU

The setup sounds almost harmless. The problem describes cattle belonging to the sun god, divided by color and sex, and gives relationships between the sizes of the groups. In mathematical form, those relationships become simultaneous Diophantine equations: equations whose solutions must be integers.

Then Archimedes makes things worse.

Additional conditions involving square and triangular numbers transform the exercise from a system that can be handled algebraically into a famously difficult number problem with an enormous smallest solution. The pastoral language is camouflage. Underneath the cows is serious number theory.

The NYU page is interesting partly because of the mathematics and partly because of how little ceremony it needs. University Web servers once accumulated thousands of pages like this: lecture notes, demonstrations, source texts, problem statements, datasets, and small research tools placed online because a faculty member or department thought somebody might need them.

They were not always designed as “content.” They were closer to shelves.

That model was extraordinarily useful. A search could land directly on the statement of a centuries-old mathematical problem without requiring an account, a viewer application, a learning platform, or a twelve-step navigation path through an institutional homepage. The URL itself points into an Archimedes directory. You are looking at the organizational habits of an earlier academic Web as well as the mathematics.

For a modern reader, the page is a useful starting point because it preserves the original shape of the challenge. You can see how an ancient word problem becomes a set of integer constraints and why the problem’s reputation is not based on merely having a lot of cattle to count.

Old university pages are fragile because they often survive outside whatever system the institution currently considers official. That makes finding and documenting them worthwhile while they are still there.

Open the Cattle Problem statement

Archimedes did not have a Web browser. He did, however, leave future computers something substantial to chew on.

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Ancient Web: KnotPlot Has Been Turning Mathematics Into 3D Knots Since the 1990s

KnotPlot looks like somebody gave topology a graphics workstation and then refused to stop experimenting for thirty years. It is part mathematical reference, part software project, part image gallery, and part personal laboratory.

Visit the KnotPlot Site

Robert Scharein built KnotPlot out of work connected to his PhD at the University of British Columbia’s Imager Computer Graphics Laboratory. The software visualizes and manipulates mathematical knots and links in three and four dimensions, and the site says almost all of its images were generated with KnotPlot itself.

That immediately makes the website more interesting than a normal gallery. The pictures are output from the subject being documented. Decorative knots, Brunnian links, torus knots, knot carpets, stereoscopic images, cubic-lattice knots, hyperbolic knots, Celtic knots, Moebius-strip knots, and stranger constructions sit alongside thesis material and technical references.

The software side is just as substantial. KnotPlot can load a database of more than 3,000 knots and links, let the user sketch knots in 3D, construct them with Conway notation, use a tangle calculator, generate special knot families, and transform existing knots. Current builds still exist for Windows, macOS, and Linux, which is remarkable enough for a program whose copyright line reaches back to 1992.

The site also retains unmistakable old-Web texture. Some galleries are polished and useful. Others are openly filed under a section titled “Pages with little or no interesting content,” because apparently deletion is for cowards. There are PostScript examples, POV-Ray output, thesis figures, stereoscopic experiments, and pages whose main justification seems to be that Scharein once wanted to see what a particular knot would look like.

That accumulation is precisely why the site matters. Modern software documentation tends to explain features. KnotPlot preserves years of intellectual wandering around the features: experiments, pictures, weird side paths, references, obsolete rendering techniques, and the research lineage behind the program.

For somebody studying mathematical visualization, early Web graphics, scientific personal sites, OpenGL software, or just the visual culture around topology, it is an unusually dense primary source.

It also demonstrates one of the Ancient Web project’s recurring lessons: a specialist website becomes valuable by refusing to flatten itself. It does not need to turn every experiment into a clean product page. The mess contains history.

More sites like this turn up in CacheRat’s 1,967 Ancient Web Domains research list, but KnotPlot stands out because the underlying software is still alive while the website continues to carry decades of accumulated research around it.

Explore KnotPlot’s galleries and software

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Ancient Web: Matthias Wandel Treated Stock Picking Like a Poker Problem

Matthias Wandel approached stock picking the way an engineer approaches a machine that everybody insists is simple right up until it eats a finger.

Read The Trouble With Investing in Stocks

The page begins with an ordinary problem: savings have accumulated, inflation exists, and leaving money idle has a cost.

Then comes the tempting part.

If the market historically grows over long periods, why settle for the market average? Why not look at charts, study companies, recognize patterns, buy low, sell high, and do better?

Wandel spends the rest of the essay worrying at that question.

Somebody has to be wrong

He first considers technical analysis: chart patterns, indicators, and the idea that price movement contains exploitable signals.

The problem, as he frames it, is that any reliable and widely recognized pattern should attract people who trade on it, which changes the pattern itself.

Value analysis does not escape much more easily. A company can look expensive right before becoming vastly more expensive, while something that looks cheap can simply continue collapsing.

Wandel eventually reaches a poker analogy.

Broad economic growth can raise the whole table over time. Beating everybody else is different. Outperformance means somebody else, somewhere, took the worse side of enough trades.

Professional managers study markets full time. Automated systems exist. Institutions have research, data, and specialists. The individual amateur who believes he has spotted the easy edge therefore has to ask an unpleasant question: who exactly is the sucker at this table?

Wandel nominates himself as a plausible candidate.

The page is not a polished financial guide and does not pretend to be one. That is what makes it interesting. It preserves a technically minded person’s reasoning process in public, including the contradictions, temptations, and skepticism.

An October 2007 update points readers to Chartgame.com, a project he created to explore pattern-based trading more directly.

CacheRat’s 1,967 Ancient Web Domains research list contains many personal pages where someone used the web as a notebook for thinking through a problem rather than as a platform for claiming expertise.

This one ends with the problem still open.

That feels more useful than a hot stock tip.

Return to Matthias Wandel’s stock-analysis essay

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Ancient Web: Pilish Turns the Digits of Pi Into English Prose

Some people memorize pi with a song. Others apparently decide English itself should be forced to carry the digits.

Visit the Pilish page

Pilish is a constrained form of writing where the length of each word encodes the successive digits of pi.

The classic example begins with words of lengths 3, 1, 4, 1, 5, 9 and so on. That works until the decimal expansion produces a zero, because ordinary prose does not contain zero-letter words.

The page’s solution is wonderfully practical: let a ten-letter word represent zero.

Writing inside a numerical cage

The site distinguishes Basic Pilish from Standard Pilish.

In Basic Pilish, words shorter than ten letters represent their literal length and ten-letter words represent zero. Standard Pilish adds a rule allowing words longer than ten letters to encode two consecutive digits. A twelve-letter word, for example, can represent 1,2.

That sounds like a tiny technical adjustment, but it gives writers much more freedom.

The page also defines how apostrophes, hyphens, punctuation, and numbers should be treated so that a text can be decoded consistently. In other words, Pilish is not merely a party trick. Somebody cared enough to write a specification.

The result is a peculiar hybrid of mathematics, mnemonics, and constrained literature.

A normal writer asks which word best fits the sentence. A Pilish writer also has to ask whether that word has exactly the right number of letters for the next digit.

That changes composition at a very low level. Syntax, vocabulary, punctuation, and even subject choice bend around the constant.

The site points to longer works built under these rules, including Cadaeic Cadenza and Not A Wake, where the constraint becomes nearly invisible unless the reader starts counting.

CacheRat’s 1,967 Ancient Web Domains research list contains many pages that preserve hobbies too specific to ever deserve a mainstream platform.

Pilish may be one of the purest examples.

It is what happens when a mnemonic becomes a literary genre.

Return to the Pilish page

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Ancient Web: Stony Brook Turns Ancient Mazes Into Mathematics

A labyrinth can be a religious symbol, a drawing exercise, an architectural pattern, or a string of numbers.

Visit Tony Phillips’ maze mathematics page

The Stony Brook page takes familiar historical labyrinths and strips them down to structure.

It compares the Cretan maze with the Jericho maze found in medieval Hebrew manuscripts, then defines a class called simple, alternating, transit mazes.

The terminology sounds dry until the diagrams make it obvious.

A maze as a level sequence

“Transit” means the path runs from the outside to the center without branches. “Alternating” means the path changes direction when it changes levels. “Simple” means it essentially circles each level once.

Once those conditions are met, a maze can be represented by the sequence of levels visited.

The page gives the Cretan pattern as 032147658 and the Jericho pattern as 03452167.

That is the lovely part: two ancient-looking drawings become objects that can be compared mathematically without caring about line thickness, decorative style, or the exact shape of the drawing.

The page also contrasts these with more complicated designs, including the Chartres-style labyrinth and a medieval plan of Constantinople, to show where the rules break.

The result is simultaneously historical and abstract. Archaeology and manuscript studies supply the examples; mathematics explains the family resemblance.

This is exactly the kind of page the old academic web was good at producing. A professor could publish a niche idea with diagrams, sequences, examples, and links without turning it into a journal article, a product, or a content campaign.

CacheRat’s 1,967 Ancient Web Domains research list repeatedly turns up academic pages whose usefulness comes from being specific enough to answer a question almost nobody else thought to ask.

This one answers a great one:

What, mathematically, makes two labyrinths the same kind of labyrinth?

Return to the Stony Brook maze page

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Ancient Web: mySundial.ca Teaches 39 Ways to Tell Time With a Shadow

There are apparently enough ways to make a shadow tell time that somebody needed an index covering thirty-nine different kinds of sundials.

Visit mySundial.ca’s Sundial Primer

Carl Sabanski’s Sundial Primer is a hand-built reference to gnomonics, the science of sundialling. The index runs through familiar horizontal and vertical dials and then keeps going: equatorial rings, analemmatic sundials, polar dials, cube dials, window dials, cylinder dials, diptychs, stained-glass dials, Foster-Lambert designs and stranger variants.

That is before the site gets to the geometry.

A clock made from astronomy

A useful sundial is more than a stick and some numbers.

The site explains latitude and longitude, longitude correction, wall declination, gnomon height, true north and south, refraction, the equation of time, solar altitude, azimuth and declination lines. It also includes construction aids, charts, templates and material for both northern and southern hemispheres.

The whole project has the unmistakable texture of specialist old-web education. There are homemade names for utilities such as the “Dialling Buddy,” “Correct-A-Dial,” “Wall Declinometer” and “SNOURTH-ometer.” It is technical material presented by somebody who clearly enjoys the fact that the subject can become ridiculous in a productive way.

The Primer advertises more than 1,400 charts and images in its extended material and paper kits for numerous sundial designs. That gives it value beyond historical curiosity. A teacher, hobbyist, garden designer or amateur astronomer can use the pages to move from the concept of solar time to a physical object.

Search engines can explain what a sundial is in one paragraph. mySundial.ca is what you find when the next question is, “Fine. How do I actually build the weird one?”

This site was found in CacheRat’s broader Ancient Web research corpus.

Explore the 1,967 Ancient Web Domains research list

The web used to be very good at letting one person build an entire small university around a narrow obsession. This is one of those places.

Return to mySundial.ca