You always should keep in mind that the BCD only measures how an object field an specific space, and not its morphology. For instance, two completely different objects could have the same BCD. If you like to know more about morphology you can complement your analysis with mass-radius dimension and Lacunarity measures.

Interesting application! But box counting may not be the best approach. Perhaps an 'oscillatory' box - a wavelet, may be a better way to address this problem.

Not to be overly self-serving but a number of people have found my book with Jim Bassingthwaighte and Larry Liebovitch, Fractal Physiology Oxford University Press, 1994) to be a useful handbook for developing methods to determine the fractal dimension of time series. There are more sophisticated texts around now, but I find they are for those that already have some experience with fractals.

Is very interedting idea, I use box counitng method in nondestructive crack atcivity monitioring heritage building and prepare other ideas for architecture and civil engineering after few years.

I used Benoit to determine box counting dimension in different Mesoamerican pyramids. An important book in this research field is: Bovil C. (1996), Fractal Geometry in Architecture and Design, Birkhauser, Boston.

You may want to search CumInCAD and also look for work from the University of Newcastle, Australia. Josephine Vaughan has published quite a few papers about fractal analysis of architectural visual complexity. You will also find other researcher's work in that area of computational analysis of architecture. http://cumincad.scix.net/cgi-bin/works/LoginForm. Searches are available to guests; access to full papers requires membership in any one of the non-profit, predominantly academic computer aided architectural design associations, ACADIA, eCAADe, CAADRIA, SIGraDi, or ASCAAD.

My question has raised several different issues in your replies : which we have been exploring in our research.

1. Measuring a fractal dimension does not necessarily mean determining if something has a fractal form. Buildings are very unlikely to ever meet the mathematical definition of a fractal, and only sometimes will they have obvious (limited) fractal aspects. We are applying the box counting method to determine the degree of visual complexity in the 2D representational image of a building, such as a plan or elevation. We are not using 3D at this stage but are interested in it for future research.

2. It is interesting to see the different programs available. We have used Benoit and now we are working on our own program. Are there any favorite features in other programs that users really appreciate?

3. Thank you for suggested references. Bruce West, I have not come across your publication yet, I will look it up. Peter Krusinsky, do you have any published works on cracks in historical buildings? Nicoletta Sala, you will find we have referenced your work in some of our publications, it is nice to hear from you directly! The works of Bovil and Eglash are very important in the area. Also thank you Volker Mueller for your suggestion to read Josephine Vaughan, I find her work to be very interesting :)

Hi Josephine! I use FracLac (thank you Audrey:) but maybe it could be interesting for you to check the Fractalyse, by professor Frankhauser and Thema from France. Also his definition of multifractals and how his Fractalyse is doing with it. Wish you all the luck with developing a new program!

Thank you Marija Jevric, I have read some of Prof. Frankhauser's work and I will have a look at Fractalyse.

In 2007, I used box-counting method for assessing the visual complexity and fractality of Hindu temples in India. At that time i used very classical Bovill's box-counting method. At present, I am working on my PhD topic for which I am learning python programming language to develop improved box-counting algorithm. Recently, I contacted to your professor M Ostwald to join his lab as a visiting research student for 1 semester and work on this project. Let's see what happens. Please, keep updating about your progress.

Josephine, I do not know how detailed your analysis should be. I mean if you consider e.g. tiny details of baroque buildings or flat curly lines of art nouveau. If you consider relatively rough structures consisting mostly from straight lines only, you can identify end points of these lines or corners in the space. Thus having set of points in three dimensional space (eventually two dimensional - photo) you can use some known procedures for stating correlation dimension D2. This is rather simplistic point of view but may be useful for your problem.

Hello Marcel,

I do not have much expertise in the computational aspect of the box counting, my basis is Architecture and Research. We are collaborating with computer engineering who have developed a program for us to calculate the fractal dimension using box counting method. The program uses line detection so we analyse quite detailed drawings and can calculate ornamental features. What we have discovered is that in some situation, diffeent levels of analysis is required. Sometimes only building footprints (eg. urban analysis) or otherwise high ornamentation (eg for a textural comaprison).

see our paper:

https://www.researchgate.net/publication/234139921_Significant_Lines_Representing_Architecture_for_Computational_Analysis?ev=prf_pub

Conference Paper Significant Lines: Representing Architecture for Computation...

Hello Josephine,

Thank you for detailed information. I read your paper with great interest. Now I would say, the work is finished as to stating the fractal dimension. I think that correlation dimension would bring no new information because results would be nearly the same and you have a good method and program already.

Hello Marcel,

Thank you for your lovely feedback! We are constantly refining our method and program and is nice to hear a knowledgeable opinion. We also feel the method and program are well considered and we are about to embark on a test of a large set of architects and buildings, and prepare to publish our research in a book.

Some considerations on MesoAmerican architecture:

http://www.generativeart.com/on/cic/papersGA2004/21.htm

and other interesting paper:

http://www.ethnomath.org/resources/burkle2001.pdf

Thank You Nicoletta Sala for those .pdf links.

I had not seen the second paper before, and it is very helpful as it does include information on the method you used, and tables of your results. Just in time for the completion of the literature review for our new book!

You always should keep in mind that the BCD only measures how an object field an specific space, and not its morphology. For instance, two completely different objects could have the same BCD. If you like to know more about morphology you can complement your analysis with mass-radius dimension and Lacunarity measures.

Thank You Roberto. Do you have any publications on this issue?

I work with fractal properties in time series data of animal behaviour. It's a long way off the string here, but some of the issues I'm interested in may also be applicable. I haven't read the various links to articles posted in this string, so I apologize if I'm being redundant, but allow me to suggest that, while calculating FD is undoubtedly a useful endeavour, perhaps equally informative could be to examine the scaling relationship underlying its calculation. So how linear is the scaling relationship? Do you see any crossover points where the scaling seems to change abruptly between 2 adjacent scales?

Not only would such information attest to the robustness of your single FD estimate, i.e. if the power-law for looks weak with points more scattered than linear the FD may have little meaning, but crossovers might also tell you something about the architecture of the structure, like that the scaling itself (the FD) depends on the range of scales examined (scale-dependence instead of scale-invariance), which may be caused by something inherent in the construction process or perhaps even patterns of wear (I'm guessing with the relevance to your field, but you can see how it's relevant to mine if you look at a recent paper we put out: https://www.researchgate.net/publication/236908211_Temporal_fractals_in_seabird_foraging_behaviour_diving_through_the_scales_of_time?ev=prf_pub). In this publication, we also compare a few different methods of fractal analysis of time series data. It may not interest you, but feel free to have a look. Oh and by the way, I have used Benoit but mainly use R packages these days (though one of which ("fractal") seems to have been dropped from CRAN due to lack of maintenance which is a bit concerning! the other is still in good standing ("fractaldim")).

Again this might not be totally relevant, but Laurent Seuront put out an impressive book on fractal analysis and its application to ecology in 2010 ("Fractals and Multifractals in Ecology and Aquatic Science"). The book discusses many methodological issues and clearly lays out the theory and approach behind many different fractal dimension and Hurst exponent estimators. For this reason, I'd recommend having a look, even if you find the examples foreign, they also serve as good illustrations of the main points.

Article Temporal fractals in seabird foraging behaviour: Diving thro...

Wow - that is fascinating stuff, thank you Andrew.

You are correct, the shape of the log-log graph of the building facades is interesting, and we have looked into it briefly, exploring if bumps are a result of the methodological variables, or a response to the details in the building facade, and how the more detailed approach to a building may affect the amount of detail immediately visible. So much more work to do in this area!

Hi Josephine,

Reff. your question to Roberto, I know that Filho and Sobreira have done some papers on urban patterns classification and they did exactly that Roberto said. They have got very similar BCD values for morphological very different patterns, so they used lacunarity as complement parameter for its classification. I have no any particular link to their papers but you can get it if you google their names + lacunarity.

Hi Marija,

Thank You for those references, and also for your endorsements for my work!

Hello Josephine:

We are working not only with the box-counting dimension but also with a generalized definition of fractal dimension that we can calculate for any subset of an space with respect to any fractal structure. So, if you change the fractal structure, the fractal dimension may also change. But if you fix the fractal dimension to consider for calculations to one very concrete, then you get the classical box-counting dimension, so you see, we work with generalized models of fractal dimension. Maybe we can help you; depending on the situation, we can change the fractal structure or the model you need. If you are interested, you can contact us in the following email: [email protected]

Regards

if you look in my profile you find box standard method as well as a very interesting new method. Visit also our site www.saistmp.com .

Hello M. Fernández-Martínez, your work sounds interesting, are you working with three dimensional fractal models?

Thank you Elio Conte, from your profile, I see you have published many papers on the subject, can you recommend one of your papers that outlines both of your box counting methods?

Hy, I think that Mr. Ion Andronache ( https://www.researchgate.net/profile/Ion_Andronache3) is interestedin fractal analysis (https://www.researchgate.net/post/Is_anyone_interested_in_applying_fractal_analysis_in_geography_hydrology_biology)

Hi, BCD is only for visual patterns, but it cannot be applied to a set which is fractal. We suggested an alternative index (ht-index) to fractal dimension for quantifying complexity of fractals or geographic features in particular:

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.

https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features

Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77.

https://www.researchgate.net/publication/270634544_Headtail_Breaks_for_Visualization_of_City_Structure_and_Dynamics?ev=prf_pub

The following paper developed a mathematical model of wholeness to quantify degree of beauty for things including visual complexity:

Jiang B. (2015), Wholeness as a hierarchical graph to capture the nature of space, International Journal of Geographical Information Science, 29(9), 1632–1648.

https://www.researchgate.net/publication/272159333_Wholeness_as_a_Hierarchical_Graph_to_Capture_the_Nature_of_Space

see the fractal patterns emerged from BIG data

Article Ht-Index for Quantifying the Fractal or Scaling Structure of...

Article Head/tail Breaks for Visualization of City Structure and Dynamics

Article Wholeness as a Hierarchical Graph to Capture the Nature of Space

As Zbigniew R. Struzik suggests, one difficulty with BC method is when trying to assess Renyi dimensions of lacunar sets. Oscillations appear in log regressions. Better to use some other tool more robust, I would say. See Physics Letters A 124(8):426-432 · 1987

Hi Josephine ,

It's really important issue that your are going to address. I wish you to address different fractal parameters such lacunarity, fractal dimensions and any other parameters that can't remember for the moment, with different techniques such as using grey level differential box , and its improvements.

This call for paper might be of your interest:

Poster CALL FOR PAPERS Fractal Geometry in Cartography and Geograph...

https://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf

Thanks Bilel Selmi, Bin Jiang's work has been very helpful in gathering an understanding of ways to understand the built environment.

Jo

Hello Josephine,

Me and my phD team (fractal studio) are still working on fractal dimension analysis using box-counting method. We have also some works with M.Ostwald about architect Sinans works (Suleymaniye & KılıcAlipasha Mosques), and lastly with Mario Lionar we do couple of work about Turkish regionalist architect Sedad Hakkı Eldem which is still goes on and printed in NNJ.

Hello Josephine, See the articles below that may help you

Jan, B., Afridi, F.A.K., Ali, M. et al. (2021). Study of the nonlinear character of ionospheric signals possessing critical frequency (foF2) at Pakistan air space. Arab J Geosci 14, 190 https://doi.org/10.1007/s12517-021-06495-8

Jan, B., Zai, M. A. K. Y., Afradi, F. K., & Aziz, Z. (2018). Study nonlinear dynamics of stratospheric ozone concentration at Pakistan Terrestrial region. Journal of Atmospheric and Solar-Terrestrial Physics, 168, 48-57

Thank you Bulbul Jan

According to the new definition of fractal (a pattern or set is fractal if the notion of far more smalls than larges recurs at least twice), the dataset [1, 1/2, 1/3, ..., 1/10] is fractal, while the dataset [1, 2, 3,..., 10] is not. For the former dataset, the notion of far more smalls than larges recurs twice, while for the latter dataset, the notion of far more smalls than larges never occurs, not to say recurs. https://en.wikipedia.org/wiki/Head/tail_breaks#/media/File:HeadTailBreaks_Classification_Illustration.png

The third definition of fractal opens a new horizon for fractal geometry:

Article New Paradigm in Mapping: A Critique on Cartography and GIS

The third definition makes fractal geometry (which is framed under the mechanistic worldview) more towards living geometry (which is framed under the organic worldview):

Article A Map Is a Living Structure with the Recurring Notion of Far...

To know more about mechanistic and organic worldviews, one can refer to my recent presentation: Presentation Spatial Healing: Bridging Space and Place through the Concep...