The Golden Section as Optical Limitation
Publication Date
July 08, 2015
Journal
PLOS ONE
Authors
Mark A. Elliott, Joy Kelly, Jonas Friedel, Jennifer Brodsky, et al
Volume
10
Issue
7
Pages
e0131045
DOI
https://dx.plos.org/10.1371/journal.pone.0131045
Publisher URL
http://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0131045
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/26154761
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4495923
Europe PMC
http://europepmc.org/abstract/MED/26154761
Web of Science
000358159700052
Scopus
84941349200
Mendeley
http://www.mendeley.com/research/golden-section-optical-limitation
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Mendeley | Further Information

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Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/2167231", "https://ndownloader.figshare.com/files/2167232", "https://ndownloader.figshare.com/files/2167233", "https://ndownloader.figshare.com/files/2167234", "https://ndownloader.figshare.com/files/2167235", "https://ndownloader.figshare.com/files/2167236", "https://ndownloader.figshare.com/files/2167237"], "description"=>"<div><p>The golden section, ϕ = (1 + √5)/2 = 1.618… and its companion ϕ = 1/ϕ = ϕ -1 = 0.618…, are irrational numbers which for centuries were believed to confer aesthetic appeal. In line with the presence of golden sectioning in natural growth patterns, recent EEG recordings show an absence of coherence between brain frequencies related by the golden ratio, suggesting the potential relevance of the golden section to brain dynamics. Using Mondrian-type patterns comprising a number of paired sections in a range of five section-section areal ratios (including golden-sectioned pairs), participants were asked to indicate as rapidly and accurately as possible the polarity (light or dark) of the smallest section in the patterns. They were also asked to independently assess the aesthetic appeal of the patterns. No preference was found for golden-sectioned patterns, while reaction times (RTs) tended to decrease overall with increasing ratio independently of each pattern’s fractal dimensionality. (Fractal dimensionality was unrelated to ratio and measured in terms of the Minkowski-Bouligand box-counting dimension). The ease of detecting the smallest section also decreased with increasing ratio, although RTs were found to be substantially slower for golden-sectioned patterns under 8-paired sectioned conditions. This was confirmed by a significant linear relationship between RT and ratio (p <.001) only when the golden-sectioned RTs were excluded [the relationship was non-significant for the full complement of ratios (p = .217)]. Image analysis revealed an absence of spatial frequencies between 4 and 8 cycles-per-degree that was exclusive to the 8-paired (golden)-sectioned patterns. The significance of this was demonstrated in a subsequent experiment by addition of uniformly distributed random noise to the patterns. This provided a uniform spatial-frequency profile for all patterns, which did not influence the decrease in RT with increasing ratio but abolished the elevated RTs to golden-sectioned patterns. This suggests that optical limitation in the form of reduced inter-neural synchronization during spatial-frequency coding may be the foundation for the perceptual effects of golden sectioning.</p></div>", "links"=>[], "tags"=>["growth patterns", "rt", "ratio", "Fractal dimensionality", "Optical Limitation", "Golden Section", "sectioned conditions", "brain frequencies", "brain dynamics", "EEG recordings show", "reaction times"], "article_id"=>1477089, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Mark A. Elliott", "Joy Kelly", "Jonas Friedel", "Jennifer Brodsky", "Paul Mulcahy"], "doi"=>["https://dx.doi.org/10.1371/journal.pone.0131045.s001", "https://dx.doi.org/10.1371/journal.pone.0131045.s002", "https://dx.doi.org/10.1371/journal.pone.0131045.s003", "https://dx.doi.org/10.1371/journal.pone.0131045.s004", "https://dx.doi.org/10.1371/journal.pone.0131045.s005", "https://dx.doi.org/10.1371/journal.pone.0131045.s006", "https://dx.doi.org/10.1371/journal.pone.0131045.s007"], "stats"=>{"downloads"=>21, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_Golden_Section_as_Optical_Limitation_/1477089", "title"=>"The Golden Section as Optical Limitation", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2015-07-08 02:49:56"}
  • {"files"=>["https://ndownloader.figshare.com/files/2167224"], "description"=>"<p>The participants’ task was to respond by button press as rapidly and accurately as possible to the luminance of the smallest section. Patterns varied in size with 4, 8 or 16 paired sections and the ratio in area of the sections with ratios of (1|0.468, 1|0.518, 1|0.568, 1|0.618, 1|0.668), where 1|0.618 is the golden section.</p>", "links"=>[], "tags"=>["growth patterns", "rt", "ratio", "Fractal dimensionality", "Optical Limitation", "Golden Section", "sectioned conditions", "brain frequencies", "brain dynamics", "EEG recordings show", "reaction times"], "article_id"=>1477082, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Mark A. Elliott", "Joy Kelly", "Jonas Friedel", "Jennifer Brodsky", "Paul Mulcahy"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0131045.g001", "stats"=>{"downloads"=>1, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Example_grids_used_in_all_experiments_from_top_to_bottom_left_to_right_a_4_paired_sectioned_pattern_with_sections_in_area_ratio_1_0_568_an_8_paired_sectioned_pattern_with_sections_in_area_ratio_1_0_618_the_golden_section_an_8_paired_sectioned_pattern_wit/1477082", "title"=>"Example grids used in all experiments, from top to bottom, left to right a 4-paired sectioned pattern with sections in area ratio 1|0.568, an 8-paired sectioned pattern with sections in area ratio 1|0.618 (the golden section), an 8-paired sectioned pattern with sections in area ratio 1|0.568 and a 16-paired sectioned pattern with sections in area ratio 1|0.668.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-07-08 02:49:56"}
  • {"files"=>["https://ndownloader.figshare.com/files/2167225"], "description"=>"<p>In (b) is shown the regression line of RTs over ratio when the golden section is excluded from analysis alongside diamonds signifying the mean RTs for all ratio conditions. A similar patterning describes RTs for Experiment 2 (b and c).</p>", "links"=>[], "tags"=>["growth patterns", "rt", "ratio", "Fractal dimensionality", "Optical Limitation", "Golden Section", "sectioned conditions", "brain frequencies", "brain dynamics", "EEG recordings show", "reaction times"], "article_id"=>1477083, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Mark A. Elliott", "Joy Kelly", "Jonas Friedel", "Jennifer Brodsky", "Paul Mulcahy"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0131045.g002", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_For_Experiment_1_a_shows_mean_RTs_and_their_standard_errors_as_a_function_of_ratio_for_the_three_pattern_sizes_4_paired_sections_diamonds_8_paired_sections_squares_and_16_pairted_sections_stars_respectively_/1477083", "title"=>"For Experiment 1 (a) shows mean RTs and their standard errors as a function of ratio for the three pattern sizes 4-paired sections (diamonds), 8-paired sections (squares) and 16-pairted sections (stars), respectively.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-07-08 02:49:56"}
  • {"files"=>["https://ndownloader.figshare.com/files/2167227"], "description"=>"<p>Box counting characterizes a fractal set by determining the number (N) of boxes of size R required to cover the fractal set, following the power law N = N0*R<sup>-df</sup> with df < = <i>d</i> (or fractal dimension). In (b) the spatial frequency structure of the golden-16-sectioned patterns, determined by application of a bank of log-Gabor filters is illustrated by the black continuous line, with golden-4- and 16-paired sectioned patterns presented for comparison purposes as gray discontinuous lines. Unlike any other pattern the 8-paired (golden) sectioned pattern exhibits an absence in spatial frequency information in the middle of the range of spatial frequencies possessed by the pattern, at around 3–7 cycles per degree of visual angle. Addition of uniformly distributed random noise (white or black pixels) to 20% of the 8-paired sectioned patterns convolved with the natural pattern amplitude spectra of the patterns to raise all lower values in the amplitude spectrum, particularly, raising zero values above zero. The black dashed line illustrates the resulting amplitude spectrum. An example pattern is given in (c); (d) shows that following this modification, in Experiment the 3, golden-section RTs were not elevated as had been found in Experiments 1 and 2 but corresponded to the approximately linear negative function describing RT over ratio.</p>", "links"=>[], "tags"=>["growth patterns", "rt", "ratio", "Fractal dimensionality", "Optical Limitation", "Golden Section", "sectioned conditions", "brain frequencies", "brain dynamics", "EEG recordings show", "reaction times"], "article_id"=>1477085, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Mark A. Elliott", "Joy Kelly", "Jonas Friedel", "Jennifer Brodsky", "Paul Mulcahy"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0131045.g003", "stats"=>{"downloads"=>0, "page_views"=>20, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_a_The_Minkowski_8211_Bouligand_or_box_counting_dimension_for_the_5_ratios_1_0_468_1_0_518_and_1_0_568_circles_squares_and_diamond_respectively_as_well_as_1_0_618_and_1_0_668_large_open_diamond_and_star_/1477085", "title"=>"(a) The Minkowski–Bouligand or box-counting dimension for the 5 ratios (1|0.468, 1|0.518 and 1|0.568 circles, squares and diamond, respectively) as well as (1|0.618, and 1|0.668, large open diamond and star).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-07-08 02:49:56"}
  • {"files"=>["https://ndownloader.figshare.com/files/2167228"], "description"=>"<p>Exponents describing the fractal dimensionality for each of the ratios employed in Experiments 1, 2 and 3.</p>", "links"=>[], "tags"=>["growth patterns", "rt", "ratio", "Fractal dimensionality", "Optical Limitation", "Golden Section", "sectioned conditions", "brain frequencies", "brain dynamics", "EEG recordings show", "reaction times"], "article_id"=>1477086, "categories"=>["Biological Sciences", "Science Policy"], "users"=>["Mark A. Elliott", "Joy Kelly", "Jonas Friedel", "Jennifer Brodsky", "Paul Mulcahy"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0131045.t001", "stats"=>{"downloads"=>5, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Exponents_describing_the_fractal_dimensionality_for_each_of_the_ratios_employed_in_Experiments_1_2_and_3_/1477086", "title"=>"Exponents describing the fractal dimensionality for each of the ratios employed in Experiments 1, 2 and 3.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-07-08 02:49:56"}

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Relative Metric

{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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