The work behind
the numbers.

Where possible we cite primary peer-reviewed sources. Books and treatises are included where the field's conceptual foundation lives there (Mandelbrot, Salingaros, Arnheim). Each citation is reproduced in standard APA form so it can be entered into reference managers without modification.

If you are using Fractune for research and need a citation for the system itself, see the citation block at the bottom of the science page.

Fractal complexity

Foundational work on fractal geometry as a descriptor of natural and designed surfaces, plus the empirical preference studies that locate the human aesthetic sweet spot near D = 1.3–1.5.

1. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W. H. Freeman.
2. Falconer, K. (2003). Fractal Geometry: Mathematical Foundations and Applications (2nd ed.). Wiley.
3. Ostwald, M.J. & Vaughan, J. (2016). The Fractal Dimension of Architecture. Birkhäuser.
4. Spehar, B., Clifford, C.W.G., Newell, B.R. & Taylor, R.P. (2003). Universal aesthetic of fractals. Computers & Graphics, 27(5), 813–820.
5. Spehar, B., Walker, N. & Taylor, R.P. (2016). Taxonomy of individual variations in aesthetic responses to fractal patterns. Frontiers in Human Neuroscience, 10, 350.
6. Taylor, R.P., Spehar, B., Wise, J.A., Clifford, C.W.G., Newell, B.R. & Martin, T.P. (2005). Perceptual and physiological responses to the visual complexity of fractal patterns. Nonlinear Dynamics, Psychology, and Life Sciences, 9(1), 89–114.
7. Taylor, R.P. (2006). Reduction of physiological stress using fractal art and architecture. Leonardo, 39(3), 245–251.
8. Allain, C. & Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44(6), 3552–3558.
9. Plotnick, R.E., Gardner, R.H., Hargrove, W.W., Prestegaard, K. & Perlmutter, M. (1996). Lacunarity analysis: A general technique for the analysis of spatial patterns. Physical Review E, 53(5), 5461–5468.
10. Mandelbrot, B.B. (1995). Measures of fractal lacunarity: Minkowski content and alternatives. In Fractal Geometry and Stochastics. Birkhäuser.
11. Karperien, A. (2013). FracLac for ImageJ. Charles Sturt University.

Edge orientation, curvature, and second-order statistics

Recent work in computational aesthetics, particularly Christoph Redies' programme, identifies edge-orientation entropy as one of the strongest single predictors of facade preference. Curvature enters via Vartanian's neuroimaging studies showing that curvilinear architectural rooms drive measurable affective responses in reward-related cortex.

12. Stanischewski, S., Altmann, C.S., Brachmann, A. & Redies, C. (2020). Aesthetic perception of line patterns: Effect of edge-orientation entropy and curvilinear shape. i-Perception, 11(5).
13. Brachmann, A. & Redies, C. (2017). High entropy of edge orientations characterizes visual artworks from diverse cultural backgrounds. Vision Research, 133, 130–144.
14. Brachmann, A. & Redies, C. (2017). Computational and experimental approaches to visual aesthetics. Frontiers in Computational Neuroscience, 11, 102.
15. Redies, C. (2015). Combining universal beauty and cultural context in a unifying model of visual aesthetic experience. Frontiers in Human Neuroscience, 9, 218.
16. Redies, C., Hasenstein, J. & Denzler, J. (2007). Fractal-like image statistics in visual art: Similarity to natural scenes. Spatial Vision, 21(1–2), 137–148.
17. Vartanian, O., Navarrete, G., Chatterjee, A., Fich, L.B., Leder, H., Modroño, C., Nadal, M., Rostrup, N. & Skov, M. (2013). Impact of contour on aesthetic judgments and approach-avoidance decisions in architecture. PNAS, 110(Suppl 2), 10446–10453.
18. Bar, M. & Neta, M. (2006). Humans prefer curved visual objects. Psychological Science, 17(8), 645–648.

Spectral statistics and rhythm

Natural scenes follow a 1/f^β amplitude spectrum with β ≈ 1.2; human visual sensitivity peaks near this slope. Fractune measures β alongside two-dimensional autocorrelation peaks for rhythm strength and fenestration regularity.

19. van der Schaaf, A. & van Hateren, J.H. (1996). Modelling the power spectra of natural images: Statistics and information. Vision Research, 36(17), 2759–2770.
20. Field, D.J. (1987). Relations between the statistics of natural images and the response properties of cortical cells. Journal of the Optical Society of America A, 4(12), 2379–2394.
21. Penacchio, O. & Wilkins, A.J. (2015). Visual discomfort and the spatial distribution of Fourier energy. Vision Research, 108, 1–8.
22. Liu, Y., Hel-Or, H., Kaplan, C.S. & Van Gool, L. (2010). Computational Symmetry in Computer Vision and Computer Graphics. Now Publishers.
23. Stamps, A.E. (2003). Advances in visual diversity and entropy. Environment and Planning B: Planning and Design, 30(3), 449–463.

Composition: balance, hierarchy, surface complexity

The architectural-composition tradition. Stamps' empirical regression identifies surface complexity as the dominant single predictor of facade preference (β = 0.72). Salingaros' theoretical framework locates "life" in hierarchical scale cooperation. Arnheim and subsequent eye-tracking work establish balance as a fundamental compositional principle.

24. Stamps, A.E. (1999). Sex, complexity, and preferences for residential facades. Perceptual and Motor Skills, 88(3c), 1301–1312.
25. Stamps, A.E. (2000). Psychology and the Aesthetics of the Built Environment. Springer.
26. Salingaros, N.A. (2006). A Theory of Architecture. Umbau-Verlag.
27. Salingaros, N.A. (2012). Fractal art and architecture reduce physiological stress. Journal of Biourbanism, 2(2), 11–28.
28. Alexander, C. (2002). The Nature of Order, Book 1: The Phenomenon of Life. Center for Environmental Structure.
29. Mehaffy, M.W. & Salingaros, N.A. (2015). Design for a Living Planet. Sustasis Press.
30. Arnheim, R. (1974). Art and Visual Perception: A Psychology of the Creative Eye. University of California Press.
31. Wilson, A. & Chatterjee, A. (2005). The assessment of preference for balance: Introducing a new test. Empirical Studies of the Arts, 23(2), 165–180.
32. Locher, P., Stappers, P.J. & Overbeeke, K. (1996). The role of balance as an organizing design principle. Perception, 27(12), 1459–1472.

Colour and naturalness

Joye's biophilic-design framework, ecological-valence theory of colour preference, and the colour-harmony tradition that informs Fractune's Chromatic Fluency C and value-contrast Δ_L.

33. Joye, Y. (2007). Architectural lessons from environmental psychology: The case of biophilic architecture. Review of General Psychology, 11(4), 305–328.
34. Kellert, S.R., Heerwagen, J. & Mador, M. (Eds.). (2008). Biophilic Design: The Theory, Science and Practice of Bringing Buildings to Life. Wiley.
35. Long, F. & Purves, D. (2003). Natural scene statistics as the universal basis of color context effects. PNAS, 100(25), 15190–15193.
36. Palmer, S.E. & Schloss, K.B. (2010). An ecological valence theory of human color preferences. PNAS, 107(19), 8877–8882.
37. Schloss, K.B. (2024). Color preferences and color harmony. Annual Review of Vision Science, 10, 323–349.
38. Itten, J. (1970). The Elements of Color. Van Nostrand Reinhold.
39. Albers, J. (1963). Interaction of Color. Yale University Press.
40. Burchett, K.E. (2002). Color harmony. Color Research & Application, 27(1), 28–31.
41. CIE (2004). Colorimetry (3rd ed.). Commission Internationale de l'Éclairage.

Architecture and the brain

The neuroscience of architectural experience, the bridge between measurable image statistics and the affective and cognitive responses architects intuit. This is also where Fractune's most important interlocutors live, including the recent Aarhus School-of-Architecture work on cognitive responses to space.

42. Coburn, A., Vartanian, O. & Chatterjee, A. (2017). Buildings, beauty, and the brain: A neuroscience of architectural experience. Journal of Cognitive Neuroscience, 29(9), 1521–1531.
43. Coburn, A., Vartanian, O., Kenett, Y.N., Nadal, M., Hartung, F., Hayn-Leichsenring, G., Navarrete, G., González-Mora, J.L. & Chatterjee, A. (2020). Psychological and neural responses to architectural interiors. Cortex, 126, 217–241.
44. Berlyne, D.E. (1971). Aesthetics and Psychobiology. Appleton-Century-Crofts.
45. Hagerhall, C.M., Purcell, T. & Taylor, R.P. (2004). Fractal dimension of landscape silhouette outlines as a predictor of landscape preference. Journal of Environmental Psychology, 24(2), 247–255.
46. Huynh, D.C. et al. (2026). Moving beyond environmental categories in environmental psychology. Journal of Environmental Psychology, 111, 103016.

Image processing primitives

Standard references for the algorithms Fractune uses for edge detection, thresholding, perspective handling, and FFT-based spectral analysis.

47. Canny, J. (1986). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(6), 679–698.
48. Otsu, N. (1979). A threshold selection method from gray-level histograms. IEEE Transactions on Systems, Man, and Cybernetics, 9(1), 62–66.
49. Hartley, R. & Zisserman, A. (2004). Multiple View Geometry in Computer Vision (2nd ed.). Cambridge University Press.
50. Wiener, N. (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. MIT Press.
51. Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379–423.
52. Karjus, A., Solà, M.C., Ohm, T., Ahnert, S.E. & Schich, M. (2023). Compression ensembles quantify aesthetic complexity and the evolution of visual art. EPJ Data Science, 12, 21.