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When people say “Gallifreyan,” they usually mean Sherman’s Circular Gallifreyan (the fan system). This one:
And buckle up mothafuckas since I'm about to dive into mathematical modelling of this beautiful bastard.
I thought about the possible word encodings and calculating info density. So, I considered symbol set sizes, entropy, error correction (checksums), redundancy, and how these relate to things like compression. I started by setting up a basic model suitable for things like Circular Gallifreyan, which might have 24 to 26 letters. I adjusted for this and worked out examples carefully.
For a word-circle with n letters, each letter occupies M angular slots, and repeated letters are allowed. The formula for capacity in bits per letter is log2(options per glyph). With 26 possible letters, each encodes as 4.7 bits. Adding features like angle or stroke variants increases options, but many are redundant and could be used for steganography. In sentences, the circle's arrangement and spacing hold extra hidden info, like covert channels.
For steganography, I considered using angle quantization by dividing the circle into bins, each hiding bits. At 1000x1000 px, the smallest discernible angle is about 1 degree, so we get around 360 bins, though realistically around 60. For letters, the angular slots would be about 3.6 bits per letter, which could support covert channels via slight jitter. I modelled literal encoding with 26 letters, yielding roughly 4.7 bits per letter.
For a word of length n, it's n * 4.7 bits without considering language redundancy. But English lets us compress further, so for large text, each character carries 1 to 1.5 bits of information. The Gallifreyan system adds redundancy for aesthetics, not cryptographic compression. I also considered a parametric system where rotations per letter increase covert capacity. For example, with 12 rotation angles, 4 dot options, and 2 thickness levels, 5 letters would give 32.9 bits of covert information.
When thinking about channel capacity under noise, I needed to consider the quantization step and potential drawing errors. To reliably transmit r bins, the bin width must be much greater than six times the standard deviation of angular error. I could also compare this to the Pigpen cipher.
Treat the script as two layers:
Literal layer (decoding-visible): a monoalphabetic substitution that maps the 26 English letters to circle+arc+dot shapes placed around a word-circle in clockwise order.
Parametric layer (reader-ignored): geometric degrees of freedom (start angle, inter-letter spacing, stroke thickness, dot count variants, radial offsets, etc.) that don’t change the decoded text but can carry extra bits (steganography).
I quantified both for Whovian's entertainment :D
1) baseline information content (literal channel)
You’re still just writing English letters in a fancy coat.
Alphabet size ≈ 26 → per-letter info = log₂(26) ≈ 4.7004 bits/char.
A word of length n: ~4.7004·n bits (before English redundancy).
Real English has ~1.0–1.5 bits/char entropy at scale, so Gallifreyan doesn’t magically compress; it just reformats.
2) where the fun starts: parametric (covert) channels
These are degrees of freedom a standard reader won’t (or isn’t supposed to) use for decoding. You can quantize them to smuggle extra bits.
Start angle of the word-circle (where the first letter sits). If you quantize to r positions (e.g., “clock hours” → r=12), capacity = log₂(r) bits/word.
Per-letter angular jitter (micro-rotation of each letter glyph relative to its nominal slot). If you allow rℓ resolvable bins per letter: log₂(rℓ) bits/letter.
Dot-count variants that the reader ignores (e.g., choose among D options that still decode the same letter): log₂(D) bits/letter.
Stroke weight choices (t thickness levels distinguishable on the medium): log₂(t) bits/letter.
Radial offset of each word inside the sentence ring (q resolvable tracks): log₂(q) bits/word.
Inter-word angular spacing (s bins for the gap size while keeping order): log₂(s) bits/gap (≈ per word, practically).
Below are the example capacities I calculated because I was bored.
Per-letter covert mix: rℓ=12 bins of micro-angle + D=4 dot options + t=2 thickness levels
→ per letter options = 12·4·2=96 → log₂(96) ≈ 6.585 bits/letter.
A 5-letter word using that mix → 5·6.585 ≈ 32.9 covert bits plus 3.585 bits from word start angle ≈ 36.5 covert bits per 5-letter word.
Sentence ring with w words and q=8 radial tracks → extra w·log₂(8) = 3w bits.
You can stack these as long as your drawing/printing medium supports the resolution reliably (more on that below).
3) design-space size (how many distinct encodings exist)
For a single word of length n:
Visible text choices (the actual word): 26ⁿ possibilities.
Covert geometry choices (example above):
per-letter 96 options · per-word start-angle 12 → 12·96ⁿ variants for the same visible word.
Total design-space for content+covert per word: 26ⁿ · 12 · 96ⁿ = 12 · (26·96)ⁿ = 12 · (2496)ⁿ encodings.
Nobody’s brute-forcing that by eyeballing a tattoo. (They don’t need to though since they’ll just read the plaintext like a normal substitution and miss the covert bits. And that’s the whole point.)
4) error model & reliable bin counts (how many bins can you actually use)
All covert capacity hinges on distinguishability under noise.
Let σθ be the standard deviation of angular placement noise (human drawing, scanning, camera skew).
For robust decoding, target bin width Δθ ≳ 6σθ (rule-of-thumb for low error).
Angular bins available on a circle: r ≈ 2π / Δθ.
If σθ ≈ 0.5° on a decent print/scan, Δθ ≈ 3°, r ≈ 360/3 = 120 bins → log₂(120) ≈ 6.91 bits for an angular parameter (per use).
Hand-drawn on skin? Maybe σθ ≈ 1.5–2°, so r ≈ 360/12 ≈ 30 bins → log₂(30) ≈ 4.91 bits. Be conservative.
Linear parameters (radial tracks, line thickness) have analogous limits: minimum resolvable step ≈ 6σ of that metric.
5) layout grammar (formal-ish view)
You can sketch a grammar to reason about degrees of freedom like so:
Total covert capacity for a sentence with w words and word lengths
Plug in the earlier example (r_word=12, rℓ=12, D=4, t=2, q=8, r_sent=12) to get concrete bit counts.
6) cryptanalysis implications
Breaking the visible text is trivial if the analyst recognizes the system (monoalphabetic substitution; frequency analysis works with enough text).
Missing the covert channel is very likely to happen. Unless an analyst hypothesizes that angles/thickness/spacing are intentional and quantized, they’ll decode the plaintext and stop. That’s covert security, not cryptographic security.
If you want actual security, layer a real cipher underneath like so:
Encrypt plaintext with Vigenère/OTP/AES → map ciphertext letters to Gallifreyan glyphs → use covert channels for MAC/check bits or extra payload.
For example, you can allocate 2–3 covert bits/letter to a parity or CRC; decoder can auto-detect tampering from hand-drawn noise.
7) practical encoding recipes
A. “clock-quantized” low-effort covert channel
Word start angle r=12 (bits/word ≈ 3.585).
Per-letter dot variant D=4 (bits/letter = 2).
Per-letter micro-angle rℓ=8 (bits/letter = 3).
Total ≈ 5 bits/letter + 3.6 bits/word.
Very robust on paper; tolerable in tattoos if you keep dots and slot angles clean.
B. “dense but fragile”
rℓ=60 (bits/letter ≈ 5.907), D=4 (2 bits), t=2 (1 bit) → ~8.9 bits/letter.
Needs print-level precision. For scans/photos, include error-correcting codes (e.g., BCH/RS) across letters.
8) steganographic throughput in images (quick sanity check)
Say you render a 1000×1000 PNG. If your minimum reliable angle step is ~1°:
r≈360 bins → 8.49 bits per angular parameter.
If you use one angular parameter per letter for a 20-letter sentence, ~170 bits hidden, plus start angles/tracks so ~200+ bits total.
That’s modest compared to LSB image stego, but here the carrier is the content itself, not the pixel noise so it is much harder to flag by generic detectors.
9) “why it works” (cognitive cheat codes)
Humans tag symmetry as art, not text so obviously there's going to be fewer attempts to decrypt. Its nonlinearity also breaks the “this is writing” heuristic, blocking casual frequency intuition. The system’s pretty geometry gives you natural side channels for hidden bits, which typical readers ignore.
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