Examining the Position of Wright’s Fallingwater in the Context of His Larger Body of Work: An Analysis Using Fractal Dimensions
Abstract
:1. Introduction
2. Background to the Research
2.1. The Positioning of Fallingwater in Wright’s Architectural Styles
2.2. The Use of Fractal Analysis to Define Architectural Style
2.3. Fractal Analysis of Frank Lloyd Wright’s Buildings
3. Method and Approach
3.1. The Box-Counting Method for Architecture
3.2. Approach
- (1)
- The fractal dimensions of the elevations of Fallingwater are developed for the first time for this paper. The fractal dimensions of three floor plans and one roof plan of Fallingwater have been recently measured and published [41]. The plans were measured using the standard architectural box-counting method [18], and the elevations are also measured in the same way in this paper. In total, eight individual D results are developed from Fallingwater for comparative purposes.
- (2)
- The primary visual characteristics of Wright’s Prairie style are considered, as in past research, to be encapsulated in a set of five key works: Robie, Evans, Zeigler, Tomek and Henderson houses. For Wright’s Textile-Block style, the Ennis, Millard, Storrer, Freeman and Lloyd-Jones houses are, for all practical purposes, the complete set. For the Usonian works, the Palmer, Dobkins, Reisley, Fawcett and Chahroudi houses are considered representative of one type of Usonian planning. Fractal dimension measures for these 15 houses (58 elevations and 46 plans) were previously produced using the standard method [18].
3.3. Derived Measures and Terminology
3.4. Interpretation of the Results
4. Results
4.1. Full Set of Fractal Dimensions for Fallingwater
4.2. Initial Comparison Fallingwater to the Prairie, Textile-Block and Usonian Sets
4.3. Detailed Comparison Fallingwater to the Prairie, Textile-Block and Usonian Sets
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Houses | Henderson | Tomek | Evans | Zeigler | Robie | Set {…} | |
---|---|---|---|---|---|---|---|
Elevations | DE1 | 1.5255 | 1.5103 | 1.5592 | 1.4442 | 1.5174 | |
DE2 | 1.5177 | 1.4885 | 1.5709 | 1.4542 | 1.5708 | ||
DE3 | 1.4910 | 1.4342 | 1.5254 | 1.4385 | 1.4785 | ||
DE4 | 1.5072 | 1.4799 | 1.5337 | 1.4424 | 1.4677 | ||
μE | 1.5104 | 1.4782 | 1.5473 | 1.4448 | 1.5086 | ||
μ{E} | 1.4979 | ||||||
M{E} | 1.4991 | ||||||
std{E} | 0.0432 | ||||||
Plans | DP-1 | 1.3001 | 1.4448 | ||||
DP0 | 1.4499 | 1.3902 | 1.4307 | 1.4170 | 1.3385 | ||
DP1 | 1.3763 | 1.3721 | 1.3817 | 1.3802 | 1.4220 | ||
DP2 | - | - | - | - | 1.3984 | ||
DPR | 1.1817 | 1.3077 | 1.3147 | 1.2295 | 1.3066 | ||
μP | 1.3270 | 1.3787 | 1.3757 | 1.3422 | 1.3664 | ||
μ{P} | 1.3579 | ||||||
M{P} | 1.3783 | ||||||
std{P} | 0.0734 | ||||||
Composite | μE+P | 1.4187 | 1.4285 | 1.4738 | 1.4009 | 1.4375 | |
Aggregate | μ{E+P} | 1.4318 |
Houses | Millard | Storer | Freeman | Ennis | Lloyd-Jones | Set{…} | |
Elevations | DE1 | 1.4420 | 1.5389 | 1.3603 | 1.6130 | 1.5947 | |
DE2 | 1.4786 | 1.5543 | 1.5125 | 1.6390 | 1.5589 | ||
DE3 | 1.3434 | 1.5111 | 1.4666 | 1.4900 | 1.6105 | ||
DE4 | 1.3128 | 1.4395 | 1.4868 | 1.4417 | 1.5983 | ||
μE | 1.3942 | 1.5110 | 1.4566 | 1.5459 | 1.5906 | ||
μ{E} | 1.4996 | ||||||
M{E} | 1.5006 | ||||||
std{E} | 0.0925 | ||||||
Plans | DP0 | 1.4078 | 1.4497 | 1.3964 | 1.4955 | 1.4465 | |
DP1 | 1.3801 | 1.4330 | 1.3799 | - | 1.4228 | ||
DP2 | 1.2826 | 1.4311 | - | - | 1.4158 | ||
DPR | 1.2809 | 1.4024 | 1.3901 | 1.4664 | 1.4127 | ||
μP | 1.3379 | 1.4291 | 1.3888 | 1.4810 | 1.4245 | ||
μ{P} | 1.4055 | ||||||
M{P} | 1.4127 | ||||||
std{P} | 0.0557 | ||||||
Composite | μE+P | 1.3660 | 1.4700 | 1.4275 | 1.5243 | 1.5075 | |
Aggregate | μ{E+P} | 1.4591 |
Houses | Palmer | Reisley | Chahroudi | Dobkins | Fawcett | Set {…} | |
---|---|---|---|---|---|---|---|
Elevations | DE1 | 1.4802 | 1.3865 | 1.4328 | 1.4596 | 1.3991 | |
DE2 | 1.4461 | 1.3710 | 1.4529 | 1.3375 | 1.5575 | ||
DE3 | 1.4642 | 1.4086 | - | 1.5359 | - | ||
DE4 | 1.4018 | 1.4265 | 1.4045 | 1.3745 | 1.4591 | ||
μE | 1.4481 | 1.3982 | 1.4301 | 1.4269 | 1.4719 | ||
μ{E} | 1.4350 | ||||||
M{E} | 1.4297 | ||||||
std{E} | 0.0560 | ||||||
Plans | DP-1 | - | 1.2968 | - | - | - | |
DP0 | 1.4412 | 1.3687 | 1.3973 | 1.3810 | 1.4155 | ||
DPR | 1.2875 | 1.3256 | 1.2908 | 1.2400 | 1.3839 | ||
μP | 1.3644 | 1.3304 | 1.3441 | 1.3105 | 1.3997 | ||
μ{P} | 1.3480 | ||||||
M{P} | 1.3687 | ||||||
std{P} | 0.0634 | ||||||
Composite | μE+P | 1.4202 | 1.3691 | 1.3957 | 1.3881 | 1.4430 | |
Aggregate | μ{E+P} | 1.4032 |
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Abbreviation | Meaning |
---|---|
D | Fractal Dimension |
DE | D for a specific elevation. |
DP | D for a specific plan. |
μE μP μE+P. | Mean D for the elevations of a building. Mean D result for the plans of a building. Mean D result for all of the plans and elevations of a building. |
μ{E} | Mean D for a set of elevations for multiple buildings. |
μ{P} | Mean D for a set of plans for multiple buildings. |
μ{E+P} | Mean D for a set of elevations and plans for multiple buildings. |
Range (%) | Qualitative Descriptors |
---|---|
x < 2.0 | ‘Indistinguishable’ |
2.0 ≤ x < 6 | ‘Very similar’ |
6 ≤ x < 11 | ‘Similar’ |
11 ≤ x < 20 | ‘Broadly comparable’ |
≥21 | ‘Unrelated’ |
Result Set | Measure | Fractal Dimension |
---|---|---|
Elevations | DE1 | 1.3321 |
DE2 | 1.4628 | |
DE3 | 1.4341 | |
DE4 | 1.3786 | |
μE | 1.4019 | |
Plans | DP0 | 1.3897 |
DP1 | 1.4439 | |
DP2 | 1.4133 | |
DPR | 1.3870 | |
μP | 1.4085 | |
Composite | μE+P | 1.4052 |
Range | RED | 0.1307 |
RE% | 13.07 | |
RPD | 0.0569 | |
RP% | 5.69 |
House Set | House 1 (Earliest House) | House 2 | House 3 | House 4 | House 5 (Latest House) |
---|---|---|---|---|---|
Prairie μE | 1.5104 | 1.4782 | 1.5473 | 1.4448 | 1.5086 |
Textile μE | 1.3942 | 1.5110 | 1.4566 | 1.5459 | 1.5906 |
Usonian μE | 1.4481 | 1.3982 | 1.4301 | 1.4269 | 1.4719 |
Fallingwater μE | 1.4019 |
House Set | House 1 (Earliest House) | House 2 | House 3 | House 4 | House 5 (Latest House) |
---|---|---|---|---|---|
Prairie μP | 1.3270 | 1.3787 | 1.3757 | 1.3422 | 1.3664 |
Textile μP | 1.3379 | 1.4291 | 1.3888 | 1.4810 | 1.4245 |
Usonian μP | 1.3644 | 1.3304 | 1.3441 | 1.3105 | 1.3997 |
Fallingwater μP | 1.4085 |
House Set | House 1 (Earliest House) | House 2 | House 3 | House 4 | House 5 (Latest House) |
---|---|---|---|---|---|
Prairie μE+P | 1.4187 | 1.4285 | 1.4738 | 1.4009 | 1.4375 |
Textile μE+P | 1.3660 | 1.4700 | 1.4275 | 1.5243 | 1.5075 |
Usonian μE+P | 1.4202 | 1.3691 | 1.3957 | 1.3881 | 1.4430 |
Fallingwater μE+P | 1.4052 |
Period | Houses | μE | μP | μE+P | Range Compared to Fallingwater | ||
---|---|---|---|---|---|---|---|
Prairie 1907–1910 | RμE% | RμP% | RμE+P% | ||||
Henderson | 1.5104 | 1.3270 | 1.4187 | 10.8450 | 8.1500 | 1.3475 | |
Tomek | 1.4782 | 1.3787 | 1.4285 | 7.6325 | 2.9800 | 2.3263 | |
Evans | 1.5473 | 1.3757 | 1.4738 | 14.5400 | 3.2800 | 6.8557 | |
Zeigler | 1.4448 | 1.3422 | 1.4009 | 4.2925 | 6.6267 | 0.4343 | |
Robie | 1.5086 | 1.3664 | 1.4375 | 10.6700 | 4.2125 | 3.2288 | |
Prairie-style set mean | 1.4979 | 1.3579 | 1.4318 | 9.6 | 2.9 | 2.6 | |
Textile-Block 1923–1929 | Millard | 1.3942 | 1.3379 | 1.3660 | 0.7700 | 7.0650 | 3.9175 |
Storer | 1.5110 | 1.4291 | 1.4700 | 10.9050 | 2.0550 | 6.4800 | |
Freeman | 1.4566 | 1.3888 | 1.4275 | 5.4650 | 1.9700 | 2.2314 | |
Ennis | 1.5459 | 1.4810 | 1.5243 | 14.4025 | 7.2450 | 11.9067 | |
Lloyd-Jones | 1.5906 | 1.4245 | 1.5075 | 18.8700 | 1.5950 | 10.2325 | |
Textile-Block-style set mean | 1.4996 | 1.4055 | 1.4591 | 9.7 | 0.3 | 5.4 | |
1937 | Fallingwater | 1.4019 | 1.4085 | 1.4052 | 0.0000 | 0.0000 | 0.0000 |
Usonian 1950–1955 | Palmer | 1.4481 | 1.3644 | 1.4202 | 4.6175 | 4.4150 | 1.4967 |
Reisley | 1.3982 | 1.3304 | 1.3691 | 0.3750 | 7.8133 | 3.6100 | |
Chahroudi | 1.4301 | 1.3441 | 1.3957 | 2.8167 | 6.4450 | 0.9540 | |
Dobkins | 1.4269 | 1.3105 | 1.3881 | 2.4975 | 9.8000 | 1.7117 | |
Fawcett | 1.4719 | 1.3997 | 1.4430 | 7.0000 | 0.8800 | 3.7820 | |
Usonian-style set mean | 1.4350 | 1.3480 | 1.4032 | 3.3 | 6.0 | 0.2 |
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Vaughan, J.; Ostwald, M.J. Examining the Position of Wright’s Fallingwater in the Context of His Larger Body of Work: An Analysis Using Fractal Dimensions. Fractal Fract. 2022, 6, 187. https://doi.org/10.3390/fractalfract6040187
Vaughan J, Ostwald MJ. Examining the Position of Wright’s Fallingwater in the Context of His Larger Body of Work: An Analysis Using Fractal Dimensions. Fractal and Fractional. 2022; 6(4):187. https://doi.org/10.3390/fractalfract6040187
Chicago/Turabian StyleVaughan, Josephine, and Michael J. Ostwald. 2022. "Examining the Position of Wright’s Fallingwater in the Context of His Larger Body of Work: An Analysis Using Fractal Dimensions" Fractal and Fractional 6, no. 4: 187. https://doi.org/10.3390/fractalfract6040187