Φ in the World of Art
Introduction
“Measure what is measurable, and make measurable what is not so.” - Galileo Galilei
Most people know the feeling of having a brilliant idea in their head, then putting pencil to paper and getting… not their idea. Everyone, even artists, struggle from this disconnect, but especially people who are not naturally artistic. In everyday life, though, this does not come up, as stick figures are usually enough to communicate ideas. But what do you do when you cannot transfer your ideas into your home? Is there some way of making interior design more quantifiable for all the math and science people?
One possibility for quantifying interior design is Φ. Φ’s mysterious attractiveness is conveyed through many famous structures and artworks, so why can it not be applied to interior design? The answer is simple: lack of recognition and practicality.
The Origins of Φ
“Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.” -Shakuntala Devi
Φ, or phi, is a mathematical constant similar to π or e. This constant, equal to 1+52, or approximately 1.618, was probably first defined by Euclid when he said, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser,” or in simpler terms, a line segment divided into two such that the ratio of the shorter segment (A) to the longer segment (B) is equivalent to the longer segment (B) to the whole line segment (A+B) (fig. 1) (qtd. in Livio).
This constant has appeared in everywhere from the swirl of a nautilus shell to some of the world’s most famous pieces of art to the Giza pyramids, which are quite possibly the oldest place Φ has been found. However, the instances of Φ in the Giza pyramids and other structures such as the pantheon appear to be happy accidents, as there is no evidence the builders consciously used Φ (Livio).
Φ in Classical Art
“Without mathematics there is no art.” - Luca Pacioli
Thus stands the 3-dimensional world, where instances of Φ’s intentional use are near non existent, but look in paintings, and it is a vastly different story. While there certainly are paintings in which there is Φ but no accompanying evidence of the painters’ evidence, as in the case of Botticelli’s The Birth of Venus, others were created with the conscious use of Φ (goldennumber.net). Leonardo da Vinci was most likely conscious of Φ, and in fact strived to incorporate the ratio into his artwork. The most prominent example of this intent is his Vitruvian Man, which was done as an illustration for “De Divina Proportione” by Luca Pacioli in 1509, making it the first time, but definitely not the last, Φ was called “the divine proportion”. Da Vinci also made use of Φ in “The Last Supper,” (fig. 2) as shown by the multitude of golden rectangles overlapping the painting (goldennumber.net).
Continuing with the last supper theme, Salvador Dali’s “The Sacrament of the Last Supper” is a highly intriguing painting, likely because of its numerous allusions to Φ. Numerous golden rectangles and golden grids can be overlayed on the painting (fig. 3). In addition, the geometric figure in the background is a dodecahedron, one of the 5 platonic solids that are closely related to Φ, dodecahedrons being perhaps the closest (Livio).
These instances of Φ are most likely being used to serve as a way to guide the viewer in feature points of the painting, as well as using Φ’s mysterious attractiveness to add to the painting. Dali’s use of the platonic solid perhaps had the intent of giving the painting a sense of benevolent mystery, the sort of which is similarly felt in some with the sense of God.
While the origins of Φ are hazy, it is demonstrated that Φ has throughout the ages been used consciously and unconsciously in the quest for beauty. From the pyramids of Giza to logos now, Φ has a way of appearing in the places least expected. If Φ can be applied to the two dimensional world so successfully, what is to say that it cannot be applied to the three dimensional world of our homes as well?
Φ in interior design
“All that is beautiful is difficult.” - Plato
Out of shelves and shelves of books on interior design, how many of them mention Φ? How many books on Φ mention interior design? How many interior design classes mention Φ? The answer is none that I’ve found. Out of all the interior design books I have perused, none of them mentioned Φ, even in a passing way. Multiple of them had sections on floor plans. Other interior design sources likewise made no mentions. Interior designers I have asked made no mention of using Φ, although my questions lacked mentioning Φ in the prompt, possibly (but not probably) skewing results. In summary, there is a distinct lack of Φ in interior design sources and vice versa.
How can Φ be incorporated in interior design? The easiest and most noticeable way to use Φ in interior design is in breaking up walls, either with artwork, shelves, or some other visual feature to form a golden rectangle, or go one step further and adjust the shelf heights to form a golden grid (fig. 4). Other ways to get a similar effect would be to alter the length of curtains hanging so a ratio of 1:Φ is achieved. A subtler method of incorporating Φ would be in floorplan. This might include using a larger piece of furniture to visually break the room into a golden rectangle, or using smaller pieces to form a vague outline of one (fig. 5). Alternately, the shape of the furniture itself could be in golden proportions, although this might require altering furniture or building it from scratch.
So why isn’t Φ used in interior design? In addition to many designers simply not being aware of Φ, it is hardly ever practical to incorporate Φ into a design. Unless the shelves are custom or you are very lucky, it can be extremely hard to find shelves that will allow you to section off a golden rectangle. Curtains, while being much easier to alter, tend to look best when they hit either right below the frame or fall nearly to the floor, not when they stop partway down the wall. Using furniture to break up the floorplan is similarly easy, but is usually not the most convenient placement of said furniture. As for buying golden furniture, unless you are buying custom furniture you will likely only find tables that are close to being golden.
Φ, Subconsciously
“Everything has beauty, but not everyone sees it.” - Confucius
Through analyzing famous paintings, it has been demonstrated that Φ shows up in a lot of places where it is unexpected. But does this trend include the average person, or is it a skill exclusive to people talented in visual arts? Assuming that my subjects were not all art proteges, it can safely be said that even the layperson’s subconscious craves Φ.
The first of the two experiments tested the rectangles people produced when prompted to draw a rectangle that looks nice them. Out of 43 results, the ratio of the rectangle closest to a golden rectangle was just 0.0254 away from being Φ. These results, configured in a box and whiskers graph (fig. 6), show a result pattern that suggests people do want to draw golden rectangles. As visually represented in the figure, the lower middle quartile is by far the densest, and is clustered around Φ. This shows that at least 25% of people surveyed drew a rectangle with a ratio similar to Φ, which I feel is significant.
The second of the experiments involved asking participants to choose a “best” and “worst” from four rectangles (fig. 7). Out of 101 participants, 48 (47.5%) chose the rectangle with a ratio of Φ as “best,” and only 7 (6.9%) chose it as “worst.” The other rectangles in the survey were a square, with 19.8% best and 23.8% worst votings, a rectangle with a ratio of φ (1/Φ) with 14.9% best and 35.6% worst votings, and a rectangle with a ratio of 1:3Φ, which received 17.8% of the best vote and 33.4% of the worst vote (fig. 8).
While the “best” vote shows a clear majority towards Φ, the “worst” vote is even more telling, with only 7 out of 101 choosing the golden rectangle as the worst. Interestingly enough, 4 of these 7 chose 3, the rectangle with a ratio of φ, equal to 1/Φ, as the best, and another 2 chose the 1:3Φ rectangle as best, with only 1 choosing the square. This could indicate that these people find other forms of Φ more attractive.
Another interesting perspective of this experiment was how few people knew about Φ, with only 12 people knowing about the golden ratio Φ, versus fraternity names or Φ as a physics symbol or greek letter. Of these 12 people, all but 2 choose the golden rectangle as best. In the previous experiment 7 out of 43 people knew of it, and in this small sample size, there appears to be a negative correlation between the knowledge of Φ in this sense and drawing a golden rectangle.
All of these results together suggest that people are indeed attracted to Φ without any prerequisite of an understanding of Φ. This agrees with findings in most historical instances of Φ, where a ratio near Φ is achieved with no apparent knowledge of Φ. It seems you do not need to know why something is beautiful to make something beautiful.
Conclusion
“A good decision is based on knowledge and not on numbers” - Plato
Simply stated, Φ does not function anywhere near as well in interior design as it does in visual art. While there is definitely Φ being used unconsciously in design, it is not to the same extent as in visual art, nor to the same effect. Perhaps this stems from the fact that paintings are meant to be seen, while houses are meant to be lived in, or perhaps it is for some reason wrapped in all that is not known about art and mathematics.
To most people Φ is just a symbol, but to others, Φ is a puzzle that might not ever be cracked, just like the mystery of what makes something beautiful. To know this would be amazing, but could knowing what makes something beautiful taint beauty? Does knowing why a flower smells so nice taint the smell of flowers? Perhaps, perhaps not. And maybe Φ is a piece of this puzzle, or maybe not, but either way, Φ has made its share of contributions to the beauty of the world, the least of which is its work in interior design.
“Measure what is measurable, and make measurable what is not so.” - Galileo Galilei
Most people know the feeling of having a brilliant idea in their head, then putting pencil to paper and getting… not their idea. Everyone, even artists, struggle from this disconnect, but especially people who are not naturally artistic. In everyday life, though, this does not come up, as stick figures are usually enough to communicate ideas. But what do you do when you cannot transfer your ideas into your home? Is there some way of making interior design more quantifiable for all the math and science people?
One possibility for quantifying interior design is Φ. Φ’s mysterious attractiveness is conveyed through many famous structures and artworks, so why can it not be applied to interior design? The answer is simple: lack of recognition and practicality.
The Origins of Φ
“Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.” -Shakuntala Devi
Φ, or phi, is a mathematical constant similar to π or e. This constant, equal to 1+52, or approximately 1.618, was probably first defined by Euclid when he said, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser,” or in simpler terms, a line segment divided into two such that the ratio of the shorter segment (A) to the longer segment (B) is equivalent to the longer segment (B) to the whole line segment (A+B) (fig. 1) (qtd. in Livio).
This constant has appeared in everywhere from the swirl of a nautilus shell to some of the world’s most famous pieces of art to the Giza pyramids, which are quite possibly the oldest place Φ has been found. However, the instances of Φ in the Giza pyramids and other structures such as the pantheon appear to be happy accidents, as there is no evidence the builders consciously used Φ (Livio).
Φ in Classical Art
“Without mathematics there is no art.” - Luca Pacioli
Thus stands the 3-dimensional world, where instances of Φ’s intentional use are near non existent, but look in paintings, and it is a vastly different story. While there certainly are paintings in which there is Φ but no accompanying evidence of the painters’ evidence, as in the case of Botticelli’s The Birth of Venus, others were created with the conscious use of Φ (goldennumber.net). Leonardo da Vinci was most likely conscious of Φ, and in fact strived to incorporate the ratio into his artwork. The most prominent example of this intent is his Vitruvian Man, which was done as an illustration for “De Divina Proportione” by Luca Pacioli in 1509, making it the first time, but definitely not the last, Φ was called “the divine proportion”. Da Vinci also made use of Φ in “The Last Supper,” (fig. 2) as shown by the multitude of golden rectangles overlapping the painting (goldennumber.net).
Continuing with the last supper theme, Salvador Dali’s “The Sacrament of the Last Supper” is a highly intriguing painting, likely because of its numerous allusions to Φ. Numerous golden rectangles and golden grids can be overlayed on the painting (fig. 3). In addition, the geometric figure in the background is a dodecahedron, one of the 5 platonic solids that are closely related to Φ, dodecahedrons being perhaps the closest (Livio).
These instances of Φ are most likely being used to serve as a way to guide the viewer in feature points of the painting, as well as using Φ’s mysterious attractiveness to add to the painting. Dali’s use of the platonic solid perhaps had the intent of giving the painting a sense of benevolent mystery, the sort of which is similarly felt in some with the sense of God.
While the origins of Φ are hazy, it is demonstrated that Φ has throughout the ages been used consciously and unconsciously in the quest for beauty. From the pyramids of Giza to logos now, Φ has a way of appearing in the places least expected. If Φ can be applied to the two dimensional world so successfully, what is to say that it cannot be applied to the three dimensional world of our homes as well?
Φ in interior design
“All that is beautiful is difficult.” - Plato
Out of shelves and shelves of books on interior design, how many of them mention Φ? How many books on Φ mention interior design? How many interior design classes mention Φ? The answer is none that I’ve found. Out of all the interior design books I have perused, none of them mentioned Φ, even in a passing way. Multiple of them had sections on floor plans. Other interior design sources likewise made no mentions. Interior designers I have asked made no mention of using Φ, although my questions lacked mentioning Φ in the prompt, possibly (but not probably) skewing results. In summary, there is a distinct lack of Φ in interior design sources and vice versa.
How can Φ be incorporated in interior design? The easiest and most noticeable way to use Φ in interior design is in breaking up walls, either with artwork, shelves, or some other visual feature to form a golden rectangle, or go one step further and adjust the shelf heights to form a golden grid (fig. 4). Other ways to get a similar effect would be to alter the length of curtains hanging so a ratio of 1:Φ is achieved. A subtler method of incorporating Φ would be in floorplan. This might include using a larger piece of furniture to visually break the room into a golden rectangle, or using smaller pieces to form a vague outline of one (fig. 5). Alternately, the shape of the furniture itself could be in golden proportions, although this might require altering furniture or building it from scratch.
So why isn’t Φ used in interior design? In addition to many designers simply not being aware of Φ, it is hardly ever practical to incorporate Φ into a design. Unless the shelves are custom or you are very lucky, it can be extremely hard to find shelves that will allow you to section off a golden rectangle. Curtains, while being much easier to alter, tend to look best when they hit either right below the frame or fall nearly to the floor, not when they stop partway down the wall. Using furniture to break up the floorplan is similarly easy, but is usually not the most convenient placement of said furniture. As for buying golden furniture, unless you are buying custom furniture you will likely only find tables that are close to being golden.
Φ, Subconsciously
“Everything has beauty, but not everyone sees it.” - Confucius
Through analyzing famous paintings, it has been demonstrated that Φ shows up in a lot of places where it is unexpected. But does this trend include the average person, or is it a skill exclusive to people talented in visual arts? Assuming that my subjects were not all art proteges, it can safely be said that even the layperson’s subconscious craves Φ.
The first of the two experiments tested the rectangles people produced when prompted to draw a rectangle that looks nice them. Out of 43 results, the ratio of the rectangle closest to a golden rectangle was just 0.0254 away from being Φ. These results, configured in a box and whiskers graph (fig. 6), show a result pattern that suggests people do want to draw golden rectangles. As visually represented in the figure, the lower middle quartile is by far the densest, and is clustered around Φ. This shows that at least 25% of people surveyed drew a rectangle with a ratio similar to Φ, which I feel is significant.
The second of the experiments involved asking participants to choose a “best” and “worst” from four rectangles (fig. 7). Out of 101 participants, 48 (47.5%) chose the rectangle with a ratio of Φ as “best,” and only 7 (6.9%) chose it as “worst.” The other rectangles in the survey were a square, with 19.8% best and 23.8% worst votings, a rectangle with a ratio of φ (1/Φ) with 14.9% best and 35.6% worst votings, and a rectangle with a ratio of 1:3Φ, which received 17.8% of the best vote and 33.4% of the worst vote (fig. 8).
While the “best” vote shows a clear majority towards Φ, the “worst” vote is even more telling, with only 7 out of 101 choosing the golden rectangle as the worst. Interestingly enough, 4 of these 7 chose 3, the rectangle with a ratio of φ, equal to 1/Φ, as the best, and another 2 chose the 1:3Φ rectangle as best, with only 1 choosing the square. This could indicate that these people find other forms of Φ more attractive.
Another interesting perspective of this experiment was how few people knew about Φ, with only 12 people knowing about the golden ratio Φ, versus fraternity names or Φ as a physics symbol or greek letter. Of these 12 people, all but 2 choose the golden rectangle as best. In the previous experiment 7 out of 43 people knew of it, and in this small sample size, there appears to be a negative correlation between the knowledge of Φ in this sense and drawing a golden rectangle.
All of these results together suggest that people are indeed attracted to Φ without any prerequisite of an understanding of Φ. This agrees with findings in most historical instances of Φ, where a ratio near Φ is achieved with no apparent knowledge of Φ. It seems you do not need to know why something is beautiful to make something beautiful.
Conclusion
“A good decision is based on knowledge and not on numbers” - Plato
Simply stated, Φ does not function anywhere near as well in interior design as it does in visual art. While there is definitely Φ being used unconsciously in design, it is not to the same extent as in visual art, nor to the same effect. Perhaps this stems from the fact that paintings are meant to be seen, while houses are meant to be lived in, or perhaps it is for some reason wrapped in all that is not known about art and mathematics.
To most people Φ is just a symbol, but to others, Φ is a puzzle that might not ever be cracked, just like the mystery of what makes something beautiful. To know this would be amazing, but could knowing what makes something beautiful taint beauty? Does knowing why a flower smells so nice taint the smell of flowers? Perhaps, perhaps not. And maybe Φ is a piece of this puzzle, or maybe not, but either way, Φ has made its share of contributions to the beauty of the world, the least of which is its work in interior design.