Araara ċ· — Fraktur

This is the thirteenth in a series of expository blog posts on Araara. In the preceding post, we introduced Apar: names for variables whose interpretation depends on an ambient type.

Let us now introduce another alphabet. We will use this alphabet to define operators, eventually building towards the hyperoperation sequence.

Recording the differences

We already know how to describe an alphabet by writing each character’s Unicode code point. But neighbouring characters often have neighbouring code points. Could we use that to shorten our descriptions?

Let’s begin by writing the first code point in Unicode o-short form: the zero-valued prefix ui, followed by the integer’s Akurto short form, then o.

Here, short form means a representation with the fewest letters, allowing both positive and negative letters. Arrange candidates in reverse Aglobasa order, then choose the alphabetically first among the shortest candidates.

For each subsequent character, we write only the difference from the preceding code point, again in Akurto short form followed by o. These differences can be positive or negative.

To recover the characters, keep a running total. The first numeral gives the first code point. Add the next difference to obtain the second, and continue.

Lowercase Fraktur

Here is our lowercase Fraktur alphabet, recorded this way:

uiwsiikvtpo to bbo peo dotao cbo jeo dbo todbo teo bbo peo dboteo dpo tco cao joao co cbo peo bo

Decoded:

𝔬 𝔞𝔢 𝔟𝔭 𝔠𝔧 𝔡𝔱 𝔣𝔳 𝔤𝔨 𝔥𝔵 𝔦𝔲 𝔩𝔯 𝔪𝔫 𝔰𝔷 𝔴𝔶

These are the lowercase Fraktur letters in Aglobasa order: twenty-five positions, with no gaps. As before, the first position is the zero element, followed by twelve inverse pairs.

Let’s decode the beginning together.

The first term, uiwsiikvtpo, gives 120108, the code point of 𝔬. Next, to contributes −14, taking us to 120094: 𝔞. Then bbo contributes four, giving 120098: 𝔢. Finally, peo contributes −3, giving 120095: 𝔟.

The spaces and line breaks are just for readability. The terminating o marks each numeral’s boundary.

Unicode gives these Fraktur letters their own code points. We are choosing different characters for our alphabet, rather than merely changing the font of ordinary Aglobasa letters. Unicode character chart

Its signature

The lowercase alphabet’s additive signature is 3,002,659. In Alungo Unicode normal form:

Ιuicaa 𐆊ssmmhhggdcc

This is the sum of all twenty-five decoded code points. The initial Attic label gives the highest level. At that level, ui contributes zero; the remaining letters give the sum.

There is a useful distinction here! Summing our delta record gives the final code point, because each difference takes us from one character to the next. To compute the alphabet’s signature, we instead sum all the running totals.

The record preserves the sequence; the signature preserves its sum.

Uppercase Fraktur

We can introduce the uppercase alphabet in the same way:

uiwslukvvo to bbo peo do
ynnrvjjo wsnlfco jeo dbo to
dbo teo bbo yzllkfo wsrrgtto
ynnrvvjo wmmlffbo tco ynnrvttco wsrrgvto
ao co yzmrvtjo wmmlfftjo bo

Decoded:

𝔒 𝔄𝔈 𝔅𝔓 ℭ𝔍 𝔇𝔗 𝔉𝔙 𝔊𝔎 ℌ𝔛 ℑ𝔘 𝔏ℜ 𝔐𝔑 𝔖ℨ 𝔚𝔜

Again, there are twenty-five positions, in Aglobasa order, with no gaps.

Some differences are much larger this time. Unicode places ℭ, ℌ, ℑ, ℜ, and ℨ in its Letterlike Symbols block, while the other uppercase Fraktur letters are in Mathematical Alphanumeric Symbols. Our record simply follows those jumps. Unicode character chart

The uppercase alphabet’s signature is 2,443,992, or, in Alungo Unicode normal form:

Ιuic 𐆊smmldba

We now have both Fraktur alphabets available, and another way to describe an ordered alphabet: write where we begin, then record how we move.

In the next post, we will use these letters to choose and combine arguments with Alocu.