Binary numbers...and others

Numbers, Regular, Binary and otherwise…

"The purpose of computing is insight, not numbers."

-Richard Hamming

I love reading about the history of mathematics. I hated doing math homework in grade school. In books on mathematical history you can usually find a blend of social and cultural history, math of course, physics, language and linguistics, and just all sorts of stuff.

One thing I do is mix numbers (1, 2, 3) and the words for those numbers (one, two, three) in my writing. I do this for clarity, although it drives a lot of people nuts. I do it to make the paragraphs look more like sentences and not scary math equations. I had a teacher that made us pick one way for paper: either type the number or write the name of the number throughout our entire essay. Well, in the immortal words of Bartleby, the Scrivener, "I'd prefer not to!"

...and on to the number systems...

The Hebrew numbering systems used 22 or sometimes 27 letters to stand for numbers, with no symbol for zero (Aleph, Bet, Gimel, etc). The Greeks had decimal numbers based on 10—but still written with letters (Alpha, Beta, Gamma, etc). Egyptians had zero, but used hieroglyphs for their numbers.

The Romans of course had their famous letter system (I, II, III, IV, etc).   They had no zero though. Yes, X stood for ‘10’ but they probably thought of it as two Vs piled on each other: five on top of five; not a ‘10’ being comprised of a one and a zero—their higher numbers carried this on because a lack of zero. But, and that’s a big but, since they set up their system based on ten (X) Roman numerals are considered to be base-10 decimal numbers. Today you can find wristwatches and clocks still using Roman Numerals.

Not only did India have zero, they had a binary mathematical system similar to Morse code and the present-day binary system computers use. Dots and dashes. Ones and zeros. Smooth or pitted hole (used by DVD and CD players). Sometimes the Indians used a symbol for zero and sometimes they left a nice space for it in their calculations, but they (and a few other cultures) recognized it just the same—and eventually treated it like any other number;  like we do today.

Eventually the base-10 numbering system we know and love today (0, 1, 2, 3, etc) came into being in what we refer to as Arabic Numerals. Although many of the ‘Arabs’ (Persians) that made it popular called it the Indian Number system—and because it wasn't binary and had a zero it was arguably better than the previously mentioned systems. Although that can be like saying "English is easier than Chinese" because that's what you may be used to.

On expensive custom watches you are sometimes given a choice of whether you want Roman numerals of ‘Arabs’.  Meaning, “do you want 1, 2, 3, etc. to be on your watch or I, IV, X, XII, etc? But that’s not the whole story:

Kisai is a wild watch manufacturer that creates watches in binary—and a bunch of other varied number system schemes. Now when someone asks you the time you can say, “I honestly don’t know—let me do some calculations and get back to you!” They’re cool looking and very thought-provoking, and give just a sample of how different ways to look at ‘numbers’ can exist in just a simple application like a wristwatch.

So, the Arab/Indian Base-10 numbering system totally rocks and is all we need right? Well, maybe if this was 1980 and you were balancing your checkbook. Today if you include what your phone, tablet, on-board car systems, computer, this blog, cable-box and TV sets utilize you may be SHOCKED to learn that the Binary Number System is what you use (whether you realize it or not) all day long, every day. Unless you’re an accountant the Binary system is what makes your modern day life function in the form of Binary Computer Code; but as I alluded to earlier: binary number systems have existed throughout the world for thousands of years!

So, what’s this Binary system in the modern sense? It’s Base-2. The two digits are (usually) a zero and a one. You've probably seen how numbers and letters and symbols on your computer keyboard are changed to a series of 1s and 0s to be understood by the computer. For example, the phrase “This is the number 73” in binary code is:

“01010100 01101000 01101001 01110011 00100000 01101001 01110011 00100000 01110100 01101000 01100101 00100000 01101110 01110101 01101101 01100010 01100101 01110010 00100000 00110111 00110011”

Okay, that’s computer coding—but what about converting a ‘regular’ (Arabic) number into binary? Do I need an online binary converter? How about an abacus or special phone app? Why don’t you just try a nice piece of paper and a pencil?

Above is the number 131 two ways: base-2 binary and base-10. For Binary you divide by 2 and write the whole number down (65) and write the remainder of 0 or 1. Once you’re done you look to the column at the right and you've got your binary 0s and 1s all lined up. Below that I did it in Base-10 to show that it works that way too! Try it in base-8 or anything else crazy you can think of.

Oh, did I say base-8 was crazy? Like 8-bits make a byte and 1024-bytes make a kilobyte. Not so crazy, huh? Although some computer applications (USB thumb drives, storage space, etc) actually use a combination scheme where only 1000 bytes make a kilobyte! Either or is potentially correct, kind of like disc vs. disk. For thousands of years there have been variations in systems that were supposed to be ‘numbers and math’, something most people think of as set in stone. Such is life sometimes. Base-8 is referred to as Octet, base-16 (which is also popular among computers and software) is called hexadecimal. There are many number systems out there. So where can you quickly see an octet and be impressed? Check your computer’s IP address. It is simply a series of four octet numbers, separated by dots:

“255.255.255.255”

By the way, I got the trick of easily converting to binary with pencil and paper from Isaac Asimov. I mean, not personally, but from his book of essays “On Numbers”. As I recall he didn't seem to remember where he got it from, and I've never heard or seen anyone else use it in my daily life.

Now you may think, “why all these systems”? Well, most non-European languages have no names for numbers much higher than 10. So, they’re open to creating systems—especially from language (alphabets)—and you get used to what you know. Also, many units of measure were based on physical phenomena like: I have ten fingers, the width of my index finger seems sort of close to the width of yours, my foot at the end of my leg marks a single pace—unless you and I have different shoe sizes, oops—people do have different shoe sizes, so older measures usually are not fantastically uniform.

Although: the seed of the Carob Tree was pretty darn uniform—so much that they used it to measure diamonds against. We still call it the ‘carat’ and it’s reasonably certain that if you picked a seed (actually a pod containing a bunch of the seeds) off a Carob Tree today it would be pretty much the same as one from a thousand years ago; and if you go to a grocery store and buy a bag of them and spill them on your kitchen floor—they’d all be the same size. Yes, you might get a weird malformed one I supposed, but the other zillion seeds on your floor would be the same size and weight as far as you could tell without using a very expensive digital scale. There are 5 (average) Carob seeds to a gram. However, between the East and West Mediterranean there actually exists a variation. One side the seeds were all one weight, the other side they were a second weight. So in the 1800s some British guys just weighed one eastern and one western seed together and divided by two. Happily it came out to 0.2 grams, which is 200 milligrams. Take five together and you get 1 gram. Isn't that convenient? It is! Just as long as there isn't a drought and you end up with acres of funny lookin' Carob seeds that is.

Variations and naming conventions make number systems confusing to most people. Take the Duodecimal system (not the Dewey Decimal system that many libraries use). You’d think the “duo” would mean two right? Nope, it’s base-12. Why on Earth would anyone want to count anything using base-12? Well, for starters, twelve hours plus another twelve hours equals a day on Earth! That’s pretty important. 

Likewise, twelve inches in a foot, but that’s the Imperial measurement system which is another can of worms entirely. Although, it just occurred to me that if a day has 24 hours it might be base-24, which is called the “quadrovigesimal” system—but it’s only called that by a very limited number of people on this planet that realize it exists. You probably shouldn't try to slip it into casual conversation.

So If you think you hate math but want to delve into things like the above, try reading some math history. 

Paper Polygons and The Rosetted Logusz Cube

Paper Polygons and The Rosetted Logusz Cube

It's The Great Stellated Dodecahedron Charlie Brown! This is the first paper polyhedron I ever made. It also got me interested in mineral crystals and magnetic ferrofluid.

Make your own here: http://www.korthalsaltes.com/model.php?name_en=great%20stellated%20dodecahedron 

A little bit boring, but necessary: Four of the Five Platonic Solids--and a hexaflexagon ring. Added my violin just to spruce up the picture. Platonic solids are 'regular' (same number of sides meating at each corner and the faces are all the same). There are only 5 of them (I made the 5th-an icosahedron-a while ago).

Trying to find out how tiny I can make inflatable superelliptiodal polyhedra. Have to use thinner paper-like drafting velum. With many polygons, particularly the Archimedean Solids:

Linear dimensions are proportional to Linear dimensions;

Area is proportional to Linear dimension squared;

Volume is proportional to Linear dimension cubed;

and you can use this knowledge to vary the sizes of your models!

One of my favorite Rubik-style puzzle: the fluctuation angle. It starts add a cube then turns into jagged star shapes. I still haven't gotten it back to cube form--or one color per side.

This all led me to burr puzzles, named because they resemble plant seed burrs.

Angel Trumpet. We get around 40 of these huge spiked pods in the garden. You cut them in halfand they have tons of seeds in them. We plant them around my house and my parents house. The flowers are big. White Angel Trumpets. Plant and they come up the following year and TAKE OVER the flowerbed worse than pachysandras, lol. I'll never have to weed the flowerbeds again.

I made this a couple days ago: it's called a 'pentagrammic antiprism'. It looks really simple/boring, but paper cutting-wise it was probably one of the most complex shapes I've cut out, taping together was difficult.

Paper geometric shapes can actually be used, not just made and forgotten about. Johann Kepler’s nested polyhedra (this one de-nested with scissors for playing on a turntable/laser pointer) showing the the cycle of Jupiter & Saturn conjunctions (positional astronomy). Circling around, every empty space between each paper column equals almost 80 years, where the columns touch the top disk equals 4 conjunctions of the planets, so about every 20 years Saturn and Jupiter conjunct. Next on is in the year 2020.

I took at piece of paper and made this stage four Koch Tetrahedron (unpacked from a stage two). I placed it next to a Bismuth crystal (one my favorite-yet cheapest possessions). I have a theory that, although very rarely occuring in crystaline form Bismuth crystals inspired the design of Pre-Columbian step pyramids. 

They used Bismuth in bronze weapon-making, and maybe on occasion they let some slag fall and cool slowly by the fire--producing these crazy irridescent crystals. Bismuth is a lot like lead (though not toxic-and even used in Pepto-Bismal). When you unpack a crystal (using math: via the Fibonacci number) or just by flipping every other level over (in paper models) you leave empty spaces: thus my two to four unpacking of the paper model--the Bismuth crystal is also hollowed out in nature (thus making it a Hopper crystal). The paper tetrahedron is about the size of my balled-up fist.

A Mayan pyramid? Iridescent man-made jewelry? Part of a circuit board? Nope! This is what happens when you heat the metallic element Bismuth until it is molten like hot Lead, and then let it slowly cool. Bismuth crystals can create stepped-pyramid shapes or boxes-within-boxes. Colors can range from mostly gold and silver chrome look, to iridescent oil-slick looking metallic blues, greens, purples and oranges.

The power and complexity of nature. Reminds me of the interior sets of the movie ‘Alien’ only more colorful.  Bismuth is used as a substitute for the metal Lead in making toy soldiers, shotgun pellets and fishing lure weights. Bismuth is also used in Pepto-Bismol! Many large bismuth crystals are made German laboratories-but it is possible to make them on a household stove-top by melting and pouring between two pans as it cools.

Ancient South/Central Americans had Bismuth-and I think it may have influenced the designs of their stepped-pyramids; but then again what do I know? This crystal is big and heavy.

Here are some I just made on my stove. The bismuth I bought was too pure: impurities and oxidation age what cause the cool colors.

All of this led up to me being interested in ferrofluid and creating my own polygon: The Rosetted Logusz Cube. Made a larger one by folding 12 isosceles triangle based oblique (non-right angled, irregular) pyramids and taping them together. The smaller one is a single piece of paper folded from a mathematical 'net'. Next will be some more permanent material.

While not maintaining their symmetry of Axes with a non-90° simple rotation-but a skewed one close to 45°, I also *inverted* some of the foldes to make the other cube disappear into the main one--instead of having the corners of both cubes poking out equally. One cube is subsumed on three corners by the other-but juts out in one corner--but also juts out in an inverted *bloom* as well: which is an inversion of a corner in the projection of an entire face of the internal (secondary?) cube. But each cube is still equal in volume~so we still have two equal cubes shown (the human eye is just drawn towards the 'main' one that still looks most 'normal'). I'm not sure-but I don't think this polyhedron has an actual name (many just have numerations of whatever they are anyway). So it's a "partially inverted-partially face tesselated compound of two cubes"...or 'The Logusz x2/45° +/- Face Rhombic' for short.

Also, it's a compound, nonconvex (and nonuniform) shape. So, not a polyhedron. Once it is stellated its lack of symmetry will be even more apparent. The solution? Interpenetrate 4 to 8 more Rosetted Logusz Cubes for symmetry! An octal-compound polyhedron!

Here's the polyhedral net for making your own:

R = Regular cube face.

O = Outward pink pyramids.

I = Inward pyramids.

A simpler way to think of this shape is to view it in its composite cell forms. Just like most stellated star-shaped polyhedra are really just an initial shape like a dodecahedron with added pyramid-shaped cells glued onto it, except that the Rosetted Logusz Cube is just a series of oblique pyramids! There is no actual center shape.

That's it! Just pyramids!

The base of each pyramid is an isosceles right triangle (90°, 45° and 45°).

The sides of each pyramid are right triangles (90°, x and y).

Here's a close-up of a triangle. If you made a bunch of pyramids out of wood the pink O pyramids and the R pyramids would just be solid pyramids. The I pyramids would be voids and actually the sides of the other pyramids, so: you would not make the I pyramids if making the Rosetted Logusz Cube out of actual solid pyramids.

The paper net needs the I pyramids because it's a hollow paper construction on a single piece of paper. You could make a bunch of paper pyramids and glue them together afterwards-you'll see you used less pyramids then the net version.

Ferrofluid DIY

Fe2O3(Co) + (CH3)2CO = Fe2O3 (and fun)!

Magnetic ferrofluid is expensive, so I decided to make my own.

Heineken mini-keg: cut the top off using a lathe; put unspooled audio tapes, unspooled VHS tapes and acetone in it; covered with foil. This is now a ferrofluid delaminator.

Iron particles sloughed off of cassette tapes via acetone bath. A simple ferrofluid showing the spikes of the magnet below. Without the magnet the particles disperse into the acetone and just sort of tint the liquid.

Place a strong magnet at the outside bottom of the keg. Gravity and occasional shaking will send the nano particles of iron to the bottom.

Every day I'd stir the tape spaghetti before and after work. I did this quickly because acetone fumes are highly inflammable and headache inducing. I kept this project outside the house!


A powerful magnet is a must. I use 1" rare-earth neodymium magnets which are actually quite dangerous. Hold one in your hand and any piece of metal can fly right into/through your flesh! These chrome looking magnets can also shatter if allowed to slam into each other. Read the reviews on Amazon about them: they're not lying about blood blisters, stitches and broken fingers.

I once stuck a 1" cube rare-earth magnet on my refrigerator--and by "stuck" I mean I had to get a pair of pliers to take it off.

This is a 1" cube rate-earth magnet. It is sinking into my bed. Not because it's heavy, it is, but because it's trying to burrow through my mattress to get to the steel support springs! It also started making my digital camera glitch so I couldn't get too close to it: notice how the sheet is a warm color but around the magnet it's greyed-out? Supposedly strong magnets cannot harm digital cameras--but something sure sucked the color right out of the center of this photo, LOL.


What's cool is that if you get a really strong magnet and some regular, non-nano sized iron fillings in mineral oil you can get the same effect!

What you are seeing is the actual shape of the magnetic field.

An easier way to see it, but in two dimensions not three is to buy a cheap piece of magnetic field viewing film which is just plastic sheets with oil and iron particles sandwiched in between.