A exploration of “Bibi-Binary” and the hexadecimal system.

A look into an interesting way of working with numbers

It has been a while since I’ve written anything concerning mathematics on this Substack, but my latest whimsical exploration into obscure areas of knowledge has prompted me to break this mathematical dry spell.

Longtime readers of this Substack may remember an essay I wrote that explored an overlooked detail in Borges’ most famous short story “Tlön, Uqbar, Orbis Tertius”, which was its many references to the Duodecimal or Base-12 numeral system. For those who don’t know or have forgotten, “Base” in this context is the term for the number of numerals a numeral-system uses. For most people the Decimal or Base-10 numeral system is the norm, which uses ten numerals (0-9) and operates in patterns of ten (ones, tens, hundreds, thousands). Since any number can be used as a base for a numeral system there are infinite possible numeral-systems, but in practice there are only a handful that are practical and useful enough for humans to bother using. Apart from the standard Decimal system, and the somewhat mathematically superior Duodecimal one, there is another that is going to be the subject of this essay, the Hexadecimal or Base-16 system.

Much like the Duodecimal system, Hexadecimal is technically superior to decimal due to having more divisors, but unlike duodecimal it has the added benefit of being a power of two. It’s this last quality that prompted its current use by many computer programmers as a more convenient way to work with binary numbers. Since it is a power of 2, any hexadecimal number can be considered a shorthand-version of a binary number. For example, the ungainly binary number “11001” (16+8+1=25) can be translated into the more economical hexadecimal number “19” (16+9=25). Since the hexadecimal numeral-system possesses 6 extra numerals, symbols beyond decimal’s 0-9 are required to express them. The standard way this is done is to use the capital letters A to F to symbolise the numerals from 10 to 15.(0123456789ABCDEF).

However, though this method is practical it is also somewhat unseemly. There’s just something inherently unaesthetic about seeing decimal numbers, such as “2025”, transformed into the alphanumeric gibberish that is the hexadecimal “7E9”. This technique also doesn’t fully reveal the system’s connection to binary, very relevant to computer programmers. Surely there’s a better way to work in Hexadecimal?

Enter Robert “Boby” Lapointe.

Lapointe was a famous pop singer and actor of 20th century France, who died in the early seventies. He was also an amateur mathematician who in his spare time decided to create the best method of working in hexadecimal thus far, the Bibi-binary system.

Lapointe’s system not only created distinct and appealing symbols and names for each of the hexadecimal numerals, but made their binary connection very explicit. The 16 symbols are divided into four groups, and each of the symbols is derived from the binary number that is its counterpart. And if for some reason the characters are inconvenient to use, one can instead express the hexadecimal numbers with their names. For instance, my iPad’s keyboard is incapable of typing 7E9 in the Bibi-binary characters, but it has no trouble typing out “BiDeKa”.

The facts I have written about Bibi-binary so far are easily found in all previously written articles on the system, of which the Wikipedia article is probably the most referenced. But I would not be writing this essay if I merely wanted to re-state what has already been said about Bibi-binary and the hexadecimal numeral system. In meditating upon and playing with this system, I have discovered aspects to it that as far as I know have not been written about previously. Most of what I have discovered about Bibi-binary was became apparent when I arranged its characters into a 4x4 grid, which revealed many interesting patterns in the characters.

Firstly, the names of the characters follow an extremely consistent order. Each row is assigned a different consonant, and each column has a vowel. The consonants symbolise the first number of their row, and the vowels represent the numbers 0-3. So “Bi” is B/4 combined with i/3 to produce 7. This simple arrangement makes it very easy to remember the order and the meaning of the Bibi-binary characters’ names.

Interestingly, the consonants “B, D, H” are the 2, 4 and 8th letters of the alphabet, all powers of 2. Only “K” breaks this pattern as it is the 11th. If Lapointe wanted to continue the pattern he should have used “P”, which is the 16th letter. I suspect that due to this break from what could have been a perfect pattern, that Lapointe was completely unaware of this hidden significance in the design of his numeral-system. The fact that the consonants aren’t in any sensible numerical order is a second clue that this is probably the case. If I were to redesign the Bibi-binary system, I would replace and reorder the consonants “H, B, K, D” with “B, D, H, P”. The vowels don’t require any modifications in my opinion, as their order seems to be start with the ones that sound the “lowest” or most “rounded”, and ends with the vowels that sound the “highest” or “sharpest”.

Secondly, another thing that is notable is the various interrelationships and amount of symmetry between the Bibi-binary characters. Look carefully at the grid I’ve drawn for about five minutes, and very quickly you will notice what I mean.

You should notice that some of the numerals, such as those for “5” and “7”, will double in value when rotated and thus become the numerals for “10”and “14”. However, the numerals for “6, 9” and “11, 13” don’t have this property. Also, some of the numerals can be derived from the combination of two numerals whose numbers add up to the first. For example, the numerals for “5” and “10” can be created by combining the ones for “4, 1” and “8, 2”, which also happen to be directly in front of and above the former two numerals.

Thirdly, it became obvious to me that due to being divided into 4 rows, the Bibi-binary system can be easily shortened into a base-4, 8 and 12 numeral system if required. As far as I’m aware no one else has noticed this possible application of the Bibi-binary system.

And fourthly, apart from creating amusing names for the hexadecimal numbers, which is Bibi-binary’s most popular feature, I’ve discovered another interesting game that can be played with Bibi-Binary. This new game is discovering which numbers produce aesthetic or symmetrical patterns. For example, “DoHoHi/B03” makes a shape very reminiscent of the horned moon symbol popular with Wiccans, and “DeDiBi/EF7” produces a nice zigzag.

All in all, Bibi-binary is a nice system that definitely deserves the rather niche interest that is gets. Though of course it could be improved. Besides my rather novel suggestions about the method of generating names for each character, many people before me have commented on the unintuitive or inconsistent way that the shape of each numeral is derived from its equivalent binary number. A blogger who wrote an essay on the hexadecimal system a year before me came up with a far superior way to do this, which you can read about here.

Finishing off this essay, I want to move on from the specific topic of Bibi-binary, to the far more general topic of hexadecimal itself. As I searched through all that I could find about hexadecimal, something that became apparent to me was that in all recorded history there has never been a human culture that exclusively used hexadecimal as its standard method of counting.

Now, most people would not find this particularly surprising since everyone knows that decimal is the “normal” way to count, and that only a few nerds with too much time on their hands would count in any way that strayed from the base of 10. But in reality, Decimal is only the most common base people use in daily life, rather than the universal default people assume it is. There are and in fact quite a few cultures that use totally different bases for their counting needs. Many tribal people in Australia and the Americas usually counted in Quinary/base-5, and some cultures developed a fondness for Vigisemal/Base-20 instead, such as the Inuits and the Mayans. There are even a few who use duodecimal, such as the Chepang of Asia. And besides this there are a handful of isolated people who took to using far more obscure numeral-systems, such as base-27, base-6 and even base-15. But not a single culture recorded in history or anthropology has used hexadecimal as its main numeral-system.

Instead, just as now, hexadecimal has had only a niche use in a handful of otherwise decimal cultures. Whereas us moderns use it exclusively for computer science, the ancient Chinese used it as the basis for most of their system of weights and measurements. And throughout Europe and the Islamic world, many occultists used it as the basis for a divination system, remarkably similar to the Chinese I Ching, called Geomancy which consisted of sixteen figures made from patterns of one or two dots. And as with duodecimal, there were a few fringe mathematicians who insisted upon replacing the decimal system with it, the most famous being the 19th century Swede John W. Nystrom.

Now, why exactly has no culture on Earth taken up Hexadecimal as its own, when even a base as ridiculous as base-27 has a handful of Papuan tribes devoted to it? To answer this, we should consider the dominant theory used to explain how humans choose a base to count with in the first place.

Pretty much all counting begins with the body. It is common knowledge that most people count decimally because they usually count with the digits of each hand. However, there are different ways that one can use the body to count, some of which imply non-decimal ways of counting. The few duodecimal-favouring cultures that exist, for example, choose to count with the 3 phalanges of each finger, totalling 12 on each hand. Vigisemal-favouring cultures instead count with the digits of the hands and feet together, giving 20 all up. Many of the more bizarre numeral-systems are found in cultures that use parts of the body besides the digits of the hands and feet, most of which seem to be located in Papua. The Base-27 favouring Telefol count from the left pinky, then move on to the other digits, arm, neck, ear and eye of the left side. Then from the nose they count the same way but on the right and in reverse, ending with the right pinky.

Armed with this information, could we perhaps find a solution to my question? Is there something about hexadecimal that precludes being easily accessible via the human body?

A fair amount of thought on my part suggests that the answer leans towards yes. It is definitely true that decimal, and to a lesser extent quinary and vigesimal, are dominant due to being the easiest to derive from the body due to being easily counted with the digits of the hands and feet. But that being said, non-decimal numeral systems can still be counted from the human body, but to do so requires a fair amount of subtlety and originality. We have already seen how some cultures derived duodecimal and base-27 numeral systems from ways of using the body that are less obvious than just using fingers and toes. After much thought, I’ve found that there are four unique ways to use the body to count in hexadecimal.

The first method is derived from the way the Yuki and Pame Indians counted in octal, or base-8. Unique among all cultures, rather than counting with the fingers, they would count the gaps between each finger and thumb, or the knuckles. If one simply applied this same method to the gaps and knuckles of the feet, 16 things to count with become available.

The second method I came up with is to combine the “normal” counting of fingers with the “Yuki” counting of the gaps between them. Count one to five “normally”, then use the thumb to count the three gaps between each finger to produce eight, or 16 with both hands.

Thirdly, one can modify the duodecimal technique of counting on the fingers by counting the tips of the fingers and the joints between the phalanges. This is actually an already established method of counting in hexadecimal that I encountered in the Wikipedia article about the subject.

Fourthly, we can go the Papuan route and count with more than just the hands. Maybe the fingers of the right hand for 1-5, then the two sections of the right arm for 6-7 and the right ear for 8, followed by the left ear for 9, the left arm for 10-11 and finally the fingers of the left hand for 12-16.

With four ways to count hexadecimally, the fact that not a single culture has used it as its default method of counting becomes even stranger. But maybe we should not assume this was or will always the case. Humanity has existed as a species for around a million years, and of that million only 10 thousand has left much in terms of archeological records. Who knows if some isolated culture in humankind’s distant and unrecorded past counted in sixteens rather than tens? And if no culture in the past ever did, then who’s to say some far-future culture 10 thousand years from now will not choose to do so instead?