Futurist49
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Five Numbers
The Dadaist movement in the early 20th century experimented with a number of techniques intended to reflect natural processes rather than intentional creative processes. This sometimes took the form of arrays of objects or shapes in a tabular format. Examples of Jean Arp’s work may be viewed at the website of the NY Museum of Modern Art (https://www.moma.org). He was unable to obtain a satisfactory image from an abstract painting he was working on, and in frustration he tore the painting into pieces and let them fall onto a background sheet; the resulting collage gave him the effect he was seeking. In later work, he developed the intention to create works based on arbitrary color selections and random arrangement of the color shapes, disconnecting the work from personal preferences of the artist. However, the method he employed could not have produced a rigorously random outcome, as I will explain below.
I chose to follow Arp’s example by creating a series of color blocks in tabular format, but using random numbers, each associated with a specific color, as the underlying design structure. The colors, intended to cover a full rainbow, were chosen from a standard student set of gouache paints.
0 - Titanium White
1 - Purple
2 - Ultramarine Blue
3 - Cerulean Blue
6 - Viridian
5 - Sap Green
4 - Lemon Yellow
7 - Yellow Orange
8 - Scarlet
9 - Crimson
Decimal Point - Lamp Black
The tables were chosen to be 12 x 9, since my 24” x 18” paper would then be easily divided into 2” x 2” squares. Thus each table contained 108 squares, 106 of which were decimals following the decimal point.
Series of random numbers can be generated using number generators found on the Internet. I chose instead to use four numbers well known to mathematicians, all of which consist of a single whole number followed by endless non-repeating digits to the right of the decimal point. Such numbers are called irrational numbers since they cannot be represented as the ratio of two whole numbers. The decimal number sequences are not truly random, as they are fixed in value according to the definition of the concept they represent; however, their behavior can be described as “pseudo-random”. The digits of pi have now been calculated beyond the ten trillionth digit, and the sequence displays the characteristics of randomness. In a truly random sequence, there is no predictable relationship between any digit and the digits that precede or follow it.
Here are the four classic constants used in this exercise, shown with their first 106 decimal digits:
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 which can be characterized in many ways. It is the base of the natural logarithms. It is the limit of (1 + 1/n)n as n approaches infinity.
e = 2.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274…….
The number π (/paɪ/; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions.
𝞹 =3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 34211706798214809...
The golden ratio in mathematics,[3] art, and architecture, known as 𝞍 (“phi”). In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
𝞍=1.618033988749894848204586834365638117720309179805762862135448622705
2604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070….
(I will admit that I made an error of omission in this table; in the top line, I left out the second 3 in the sequence….803398…. I’m pretty sure that the math nerds on this site would have pointed it out to me!)
The reciprocal Fibonacci constant, or ψ (“psi”), defined as the sum of the reciprocals of the Fibonacci numbers.
Numerically, ψ = 3.35988566624317755317201130291892717968890513373196 8486495553 81532513031899668338361541621645679008729704534292885391330413678901710088367959 1351733077….
The fifth table represents a series developed by my choice. The intention was to create a sequence that “looked” random, spreading the colors out by eye so that they were more or less evenly distributed. I also specified that the colors would be shown in more or less equal amounts, appearing 10 or 11 times in the table. I call this a “haphazard”, rather than random, distribution. In the other four tables, the numbers were generated and the colors placed accordingly; in this one the colors were first placed, and the number was decoded from the color chart.
The number arrived at was:
5.8921029 213795813 849543847 357620910 128014253 379657971 854138468 957290752 264830301 039357389 453170134 780392015
I know of no special significance to this number.
Some Dada tables are very clean and formal, with neatly cut squares and smooth uniform colors. I chose instead to allow for loose brushwork, with variations in depth of color, and ragged edges to the squares. This to me seemed a better approach, giving a more spontaneous, organic look to the work.
Inspection of the resulting tables reveals immediately a key difference between random and haphazard arrangements. Some colors (numbers) appear more frequently than others in the pseudo-random tables, whereas in the haphazard table they are, by choice, equally represented. Notably, there is much more clustering (same colors side by side) and grouping (colors arranged in numerical order) in the pseudo-random sequences than in the haphazard sequence. A characteristic of random number sequences is that they will produce every possible cluster and every possible group, if the sequence is run out long enough. I attempted not to place two blocks of the same color side by side, thinking that it would look more random. However, in my attempt to create randomness, I actually imposed constraints upon the table which produced a less random result. Whether it looks prettier than the others, or not, is a matter of personal taste.
My conclusion from this is that Jean Arp’s attempts at randomness tended to restrict the outcomes and impose unintentional discipline upon the final form of the work. My four depictions of pseudo-random sequences suggest that apparently random events may actually be signs of a deeper structure by which the numbers in the sequence are related. In all four cases, the number sequence is derived from the expression of a single mathematical concept. It is impossible to predict the next number in the sequence by examining the sequence itself; however, the next digit can be mathematically calculated using known techniques. In the haphazard sequence, I recognise that intentional choices, whether unconscious or conscious, can never be truly random, since our intuition rejects clustering and grouping as non-random, whereas the math assures us that they are an inevitable consequence of truly random events.
The Dadaist movement in the early 20th century experimented with a number of techniques intended to reflect natural processes rather than intentional creative processes. This sometimes took the form of arrays of objects or shapes in a tabular format. Examples of Jean Arp’s work may be viewed at the website of the NY Museum of Modern Art (https://www.moma.org). He was unable to obtain a satisfactory image from an abstract painting he was working on, and in frustration he tore the painting into pieces and let them fall onto a background sheet; the resulting collage gave him the effect he was seeking. In later work, he developed the intention to create works based on arbitrary color selections and random arrangement of the color shapes, disconnecting the work from personal preferences of the artist. However, the method he employed could not have produced a rigorously random outcome, as I will explain below.
I chose to follow Arp’s example by creating a series of color blocks in tabular format, but using random numbers, each associated with a specific color, as the underlying design structure. The colors, intended to cover a full rainbow, were chosen from a standard student set of gouache paints.
0 - Titanium White
1 - Purple
2 - Ultramarine Blue
3 - Cerulean Blue
6 - Viridian
5 - Sap Green
4 - Lemon Yellow
7 - Yellow Orange
8 - Scarlet
9 - Crimson
Decimal Point - Lamp Black
The tables were chosen to be 12 x 9, since my 24” x 18” paper would then be easily divided into 2” x 2” squares. Thus each table contained 108 squares, 106 of which were decimals following the decimal point.
Series of random numbers can be generated using number generators found on the Internet. I chose instead to use four numbers well known to mathematicians, all of which consist of a single whole number followed by endless non-repeating digits to the right of the decimal point. Such numbers are called irrational numbers since they cannot be represented as the ratio of two whole numbers. The decimal number sequences are not truly random, as they are fixed in value according to the definition of the concept they represent; however, their behavior can be described as “pseudo-random”. The digits of pi have now been calculated beyond the ten trillionth digit, and the sequence displays the characteristics of randomness. In a truly random sequence, there is no predictable relationship between any digit and the digits that precede or follow it.
Here are the four classic constants used in this exercise, shown with their first 106 decimal digits:
The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 which can be characterized in many ways. It is the base of the natural logarithms. It is the limit of (1 + 1/n)n as n approaches infinity.
e = 2.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274…….
The number π (/paɪ/; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions.
𝞹 =3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 34211706798214809...
The golden ratio in mathematics,[3] art, and architecture, known as 𝞍 (“phi”). In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
𝞍=1.618033988749894848204586834365638117720309179805762862135448622705
2604628189024497072072041893911374847540880753868917521266338622235369317931800607667263544333890865959395829056383226613199282902678806752087668925017116962070….
(I will admit that I made an error of omission in this table; in the top line, I left out the second 3 in the sequence….803398…. I’m pretty sure that the math nerds on this site would have pointed it out to me!)
The reciprocal Fibonacci constant, or ψ (“psi”), defined as the sum of the reciprocals of the Fibonacci numbers.
Numerically, ψ = 3.35988566624317755317201130291892717968890513373196 8486495553 81532513031899668338361541621645679008729704534292885391330413678901710088367959 1351733077….
The fifth table represents a series developed by my choice. The intention was to create a sequence that “looked” random, spreading the colors out by eye so that they were more or less evenly distributed. I also specified that the colors would be shown in more or less equal amounts, appearing 10 or 11 times in the table. I call this a “haphazard”, rather than random, distribution. In the other four tables, the numbers were generated and the colors placed accordingly; in this one the colors were first placed, and the number was decoded from the color chart.
The number arrived at was:
5.8921029 213795813 849543847 357620910 128014253 379657971 854138468 957290752 264830301 039357389 453170134 780392015
I know of no special significance to this number.
Some Dada tables are very clean and formal, with neatly cut squares and smooth uniform colors. I chose instead to allow for loose brushwork, with variations in depth of color, and ragged edges to the squares. This to me seemed a better approach, giving a more spontaneous, organic look to the work.
Inspection of the resulting tables reveals immediately a key difference between random and haphazard arrangements. Some colors (numbers) appear more frequently than others in the pseudo-random tables, whereas in the haphazard table they are, by choice, equally represented. Notably, there is much more clustering (same colors side by side) and grouping (colors arranged in numerical order) in the pseudo-random sequences than in the haphazard sequence. A characteristic of random number sequences is that they will produce every possible cluster and every possible group, if the sequence is run out long enough. I attempted not to place two blocks of the same color side by side, thinking that it would look more random. However, in my attempt to create randomness, I actually imposed constraints upon the table which produced a less random result. Whether it looks prettier than the others, or not, is a matter of personal taste.
My conclusion from this is that Jean Arp’s attempts at randomness tended to restrict the outcomes and impose unintentional discipline upon the final form of the work. My four depictions of pseudo-random sequences suggest that apparently random events may actually be signs of a deeper structure by which the numbers in the sequence are related. In all four cases, the number sequence is derived from the expression of a single mathematical concept. It is impossible to predict the next number in the sequence by examining the sequence itself; however, the next digit can be mathematically calculated using known techniques. In the haphazard sequence, I recognise that intentional choices, whether unconscious or conscious, can never be truly random, since our intuition rejects clustering and grouping as non-random, whereas the math assures us that they are an inevitable consequence of truly random events.
