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Fractal geometry of literature: first attempt to Shakespeare's works

Paper · 2004 · Ali Eftekhari

Summary

This 2004 paper asks whether written literature has measurable geometric order. The author adapts a fractal-dimension method from music analysis by Hsu and Hsu. In that method, the frequency of pitch intervals follows an inverse power law. Here, alphabet letters take the place of musical notes. Each letter gets an index equal to its alphabetical position, with an imaginary zero-index letter placed before A. Letter counts for ten Shakespeare tragedies are fitted on a log-log scale, and the slope is read as a fractal dimension. The fit is very poor. The author argues that the correlation coefficient itself can be read as a "degree of fractality," and that literature is simply weakly fractal.

The paper then applies Zipf's law to letters rather than words. It defines a "Zipf's order" (letters ranked by frequency) and a "Zipf's dimension" (the fitted exponent). Both dimensions move together and are larger for shorter texts. A frequency-ranked log-log plot fits much better than semi-log plots. The claimed value is a language-independent way to analyse texts.

Key points

  • Hamlet's letter counts plotted in alphabetical order give a fractal dimension of 0.45, but with a very weak fit (R² ≈ 0.067).
  • The data set is ten Shakespeare tragedies, each reduced to the counts of the 26 letters.
  • When letters are ranked by frequency, the Zipf's order is nearly identical across all the tragedies. The author links this to roughly stable letter frequencies in English.
  • For frequency-ranked Hamlet data, three plot forms were compared by correlation coefficient:
    • P vs log(i): ≈ 0.675
    • log(P) vs i: ≈ 0.847
    • log(P) vs log(i): ≈ 0.955, the best, which points to a power law.
  • Fractal dimension and Zipf's dimension track each other, and both shrink as manuscripts get longer. There is one exception near 100,000 letters.

Interesting ideas

  • I1. Report a fitted parameter together with its goodness of fit as a separate quantity, treating a weak fit as information about the object rather than discarding it.
  • I2. Whether a lawful pattern shows up depends on the ordering chosen: alphabetical ordering looked chaotic, while frequency ranking became monotonic and fit well.
  • I3. Pick a model form by fitting several transforms (linear, semi-log, log-log) and comparing correlation, rather than assuming one in advance.
  • I4. Summary statistics of texts drift with text length, so comparisons across documents need length control.
  • I5. The author presents applicability "even to random texts" as a strength. A statistic that cannot separate meaningful text from noise, however, says little about meaning.

For Mova

How these ideas could serve Mova, our small open language model — its files, stages and tests:

  • I1 (learn): Evaluation dashboards could show both a metric and how well a rule or model actually fits the data. This would flag scores that rest on shaky regularities.
  • I2 (tokens): When profiling tagger or lemma error distributions, try several orderings (by frequency, by class, by position) before concluding there is no structure.
  • I3 (learn): A cheap model-form check could be applied to learning curves or treebank-size vs LAS relations, to see which scaling law actually holds.
  • I4 (answer): When comparing difficulty or style measures across fables, school texts and word problems, normalise or bucket by length so length effects are not mistaken for content.
  • I5 (learn): Any corpus-level diagnostic Mova adopts should be tested against shuffled or random-text negative controls, to confirm it reflects structure rather than mere letter or word frequencies.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Fields, topics and concepts

📚 Shelves 6The field of science this work belongs to (OpenAlex fields).Arts and Humanities · 16,039 worksBiochemistry, Genetics and Molecular Biology · 234 worksMolecular Biology · 231 worksLiterature and Literary Theory · 519 worksEconomics, Econometrics and Finance · 17 worksEconomics and Econometrics · 12 works

Cite

Cite as Ali Eftekhari. 2004. Fractal geometry of literature: first attempt to Shakespeare's works. arXiv:cs/0408041.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.cs/0408041 · arXiv cs/0408041

@misc{eftekhari2004fractal,
  title = {{Fractal geometry of literature: first attempt to Shakespeare's works}},
  author = {Ali Eftekhari},
  year = {2004},
  doi = {10.48550/arxiv.cs/0408041},
  eprint = {cs/0408041},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.cs/0408041}
}

References

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  35. R. F. Cancho and R. V. Sole, Adv. Complex Sys. 5, 1 (2002). Table 1. Characteristics data for letters applied in the Hamlet manuscript. Letter Letter interval (i) Incidence Incidence % (F) [a] Base letter 0 0 0 A 1 10,251 7.5931645963423 B 2 1,816 1.34515529284534 C 3 2,840 2.10365695577135 D 4 5,375 3.98139300608135 E 5 15,845 11.7367762197877 F 6 2,712 2.0088442479056 G 7 2,493 1.84662563054154 H 8 8,639 6.399117056658 I 9 8,905 6.59614971519152 J 10 110 0.0814796708221299 K 11 1,257 0.931090420212884 L 12 6,489 4.80655985422546 M 13 4,239 3.1399302237728 N 14 8,578 6.35393287556573 O 15 11,450 8.48129300830352 P 16 2,006 1.48589290608357 Q 17 218 0.161477893083857 R 18 8,100 5.99986666962956 S 19 8,668 6.42059806078384 T 20 12,450 9.2220172885047 U 21 4,738 3.50955163959319 V 22 1,219 0.902942897565239 W 23 3,110 2.30365251142567 X 24 177 0.131108197595609 Y 25 3,198 2.36883624808338 Z 26 120 0.0888869136241417 [a] This is an assumptive character introduced for calculating the value of I for 26 letters of English alphabets. All intervals were estimated in accordance with this assumptive letter, but it has no role in the statistical analysis of the literature. Table 2. Calculated values for the fractal dimensions of different tragedies of Shakespeare. Tragedies Number of total letters 116,209 Anthony and Cleopatra Coriolanus Hamlet Julius Caesar King Lear Macbeth Othello Romeo and Juliet Timon of Athenes Titus Andranicus 124,626 135,003 86,659 115,986 77,524 115,245 105,834 83,500 92,467 Fractal dimension (D f ) 0.5516 Correlation coefficient (R 2 ) 0.07923 0.4707 0.4500 0.5269 0.5598 0.5985 0.5699 0.5496 0.5358 0.5180 0.06009 0.06712 0.07023 0.07934 0.09261 0.08031 0.08686 0.07680 0.06743 Table 3. Analysis of different Shakespeare’s tragedies based on Zipf’s law. Zipf slope [a] Tragedies Anthony and 0.40587 1.9320 Cleopatra Coriolanus 0.40876 0.927299 1.8764 0.955026 Hamlet 0.40585 0.928312 1.6973 0.954612 Julius Caesar 0.40596 0.948025 1.9448 0.956628 King Lear 0.40276 0.913496 1.9424 0.955115 Macbeth 0.40247 0.928162 1.9414 0.974379 Othello 0.41361 0.925255 1.9720 0.961605 Romeo and Juliet 0.40588 0.920359 1.8441 0.977931 Timon of Athenes 0.41199 0.927070 1.8935 0.959367 Titus Andranicus 0.40869 0.936204 1.9558 0.961202 [a] Slope of the F (percent of incidence) – Letters (arranged in Zipf’s order) plot. [b] Estimated from the slope of the log (F) – log (1/n) plot, where n refers to the letter in Zipf’s order. Indeed, D Z is the exponent of a in the Zipf’s equation. Correlation coefficient (R 2 ) 0.921137 Zipf dimension [b] (D ) Z Correlation coefficient (R 2 ) 0.972011 Figure Legends: Fig. 1. Appearance of letters in a typical Shakespeare’s work, a famous tragedy; Hamlet. Fig. 2. Plotting F against 1/i in a log-log scale for a typical Shakespeare’s work, the Hamlet manuscript. Fig. 3. Degree of fractality presented for a series of Sierpinski gaskets. Fig. 4. The numbers of each letters repeated in the Hamlet tragedy as a function of letters ordered due to their intervals in Zipf's order. Fig. 5. Zipf’s order of letters for different Shakespeare’s tragedies. Fig. 6. Variations of both Zipf’s dimension D Z and fractal dimension D F (fractal dimension was illustrated one unit higher as D F +1 for comparison purpose) for different manuscripts. The manuscript# is referred to different Shakespeare’s works investigated in this research arranged in accordance with their lengths (number of letters). Fig. 7. Plotting the data obtained from Hamlet manuscript in different forms: (a) P – log (i), (b) log (P) – i, (c) log (P) – log(i). The percent of incidence (P) was obtained from the data reported in Table 1 and the rank i is the Zipf’s order reported in Fig. 5. Figure 1 Figure 2 Figure 3. Figure 4. Figure 5. Figure 6. Figure 7(a). Figure 7(b). Figure 7(c). find

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