Celestial Calendar
The Celestial Calendar is an interesting, and quite complex, constructed calendar I found while looking through the website for Jaobon, a conlang by the same creator. It was made with the stated intent of looking and feeling like traditional calendars from around the world, and though I'm not sure if it succeeds in that regard, I find the geometry and symmetry of it very pleasing.
The calendrical system itself isn't too complicated. A year contains 4 seasons; a season contains 3 months; a month contains 5 weeks; and a week contains 6 days. The remaining 5 to 6 days of the year are incorporated as New Year's Day, Leap Day (when applicable), and four holidays drawn from Celtic paganism: Imbolc, Beltane, Lunasa, and Samhain. It uses solar seasons - winter is the quarter with the least sunlight, summer the quarter with the most, and spring and autumn fit between. This means the solstices and equinoxes fall in the middle of a season, which pleases me greatly. I also like that the months are all exactly the same length.
The years are further grouped into cycles, then ages. The calendar begins somewhat arbitrarily on the Gregorian year of 3346 BCE, approximately the date of the first known writing.
Layered over this timekeeping system is a complex set of associations inspired by those of tradiitonal Western and Chinese elements. The months are associated with celestial symbols (suns, moons, or stars). (They get their names from animals, but the creator says this is "provisional".) Each day of the week is associated with a color, element, and celestial body, which gives the day its name. Each element is itself associated with a celestial symbol,season, and direction, leading to a very complex system of symbolic connections.
Rather than a continual count from Year 0, like the Gregorian Calendar, the Celestial Calendar expresses the time from Year 0 as a year-cycle-age arrangement, e.g. 2026 is the 124th year of the 6th cycle of the 4th age. It's hard for me to imagine this being used in daily life, so I modestly propose that, in daily life, the year of an occurence could simply be expressed as the number of years since Year 0, e.g. I'm writing this in 5,372, that is, 2026. Or just chain them together: 124-6-4.
Swatch Internet Time
In 1998, the Swatch corporation introduced Swatch Internet Time, a decimal time system intended for use online, alongside their line of .beat watches, which displayed SIT time alongside standard time. Decimal time is a system where time is measured in multiples of ten, that is, decimally! SIT, also known as .beat time, divides a day into 1,000 .beats, each lasting 86.4 seconds or 1.44 minutes. To avoid confusion between people in different time zones (which can be common online), SIT has no time zones and does not observe Daylight Savings: @500 BEATS is @500 BEATS for everyone. A day in SIT turns over at midnight in Biel/Bienne, a city in Switzerland where Swatch is headquartered. @248 is is 248 beats after midnight in UTC+01:00, approximately 5:57 AM.
SIT in its original implementation had no larger or smaller units equivalent to hours or seconds, though some third parties using .beat time have introduced smaller units.
Base 6 (Seximal)
Most of the modern world uses a base 10 positional numbering system. That is to say: we use a number system where each digit in a number represents a quantity of multiples of ten. Consider the decimal numeral 3125:
- The leftmost digit represents the number of thousands. 3000 is 3 (the written digit) times 10^3, or 1000. Notice how the 10^3 place has 3 digits to the right.
- The next represents the number of hundreds. 100 is 1 times 10^2 (100).
- The next is the number of tens. 20 is 2 times 10^1 (10).
- The rightmost digit is the number of ones. 5 is 5 times 10^0 (1).
It's base 10 because every position represents the number of a power of ten. It's positional because the position of the digit indicates its value (3 vs. 30 vs. 300).
Base 6 positional numbering is therefore a system where each digit of a number represents some multiple of a power of 6! 6^1 is 6, and 6^0 is still 1, so 14 in base 6 is equivalent to 10 in base 10 (1*6 + 4*1). 100 in base 6 is 36 in base 10 (6^2).
Base 6 has some really interesting aspects. Because 6 is a multiple of 3 and 2, any power or multiple of six is also evenly divisible by 3 and 2. If you've ever been annoyed by how multiples of 10 always go into infinite decimals when divided by 3, base 6 does not have that problem. It's a small enough number to be easily counted on one's hands. You don't need to invent new symbols or co-opt existing symbols to write in base 6, unlike base 12, another popular alternative base, and it's not so small that common numbers become intolerably long (such as with binary, base 2).
I'm personally not enough of a math nerd to be extremely enthusiastic about base 6, though I think it's pretty cool. I was introduced to it via jan Misali (the Conlang Critic) making two videos and a website about it, which I recommend looking at if you find the idea of base 6 interesting!
Quikscript
This is more linguistic than mathematical, but I couldn't come up with a better name for this page and I didn't want to make a second one. Quikscript is a new alphabet for the English language, developed by Ronald Kingsley Read in 1966. The core idea of Quikscript (and its predecessor, the Shavian alphabet) is that it's entirely phonetic - that is, each Quikscript letter corresponds to one sound, and words are written exactly how they're said. Read was frustrated with how English is written in the Latin alphabet, because of the often convoluted spellings, but also because it was slow - Shavian and Quikscript were both heavily based on Pitman shorthand. I have a soft spot for English spelling reforms, especially those that aim to be entirely phonetic, and I actually keep a diary in Quikscript. I do really wish it had capitals, though, and less tall and deep letters.
An interesting complication of phonetically spelling English is that English has a huge variety of dialects with unique sound inventories. Quikscript is specifically written with British English in mind, with small allowances for Scottish English and Welsh. A lot of other English dialects have sounds that aren't accounted for in Quikscript, and I don't feel wise enough to suggest how to manage that.
If you're interested in learning, or learning more about, Quikscript, friedorange.xyz has plenty of guides, including links to Read's original Quikscript manual.