Braces Of Eight
Digital Print 2021The number eight is shown using the set-based construction of the non-negative integers described by John von Neumann in 1925. Zero is represented by the empty set {} and the successor of an integer n is the union of the set n with the set containing n.
Background and Inspiration
How does one define the concept of number? What is the meaning of a number such as eight? One way to do this is from the perspective of a set.
In this set-theoretic finite ordinal number representation, zero is represented by the empty set: \(\{\}\). Starting with \(0 = \{\}\), we can define the integers recursively with the use of a successor function, \(s(n)\), defined as follows: \[s(n) = n + 1 = n \cup \{n\}.\] The successor of 0, \(s(0)\), is 1: \[1 = s(0) = 0 \cup \{0\} = \{\} \cup \{\{\}\} = \{\{\}\}.\] The successor of 1, \(s(1)\), is 2: \[2 = s(1) = 1 \cup \{1\} = \{\{\}\} \cup \{\{\{\}\}\} = \{ \{\}, \{\{\}\}\}.\] The successor of 2, \(s(2)\), is 3: \[\begin{aligned} 3 = s(2) = 2 \cup \{2\} &= \{ \{\}, \{\{\}\}\} \cup \{\{ \{\}, \{\{\}\}\}\} \\ &= \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}.\\ \end{aligned}\] The successor of 3, \(s(3)\), is 4: \[\begin{aligned} 4 &= s(3) \\ &= 3 \cup \{3\} \\ &= \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\} \cup \{\{\{\},\{\{\}\}, \{\{\},\{\{\}\}\}\}\} \\ &= \{\{\}, \{\{\}\}, \{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\}.\\ \end{aligned}\] In general, the number \(n\) is the successor of \(n-1\) and is the set containing the numbers \(0\) through \(n-1\): \[\begin{aligned} s(n-1) &= (n-1) \cup \{(n-1)\}\\ &= \underbrace{(n-2) \cup \{(n-2)\}}_{(n-1)} \cup \{(n-1)\}\\ &= \underbrace{(n-3) \cup \{(n-3)\}}_{(n-2)} \cup \{(n-2)\} \cup \{(n-1)\}\\ &= \cdots\\ &= 0 \cup \{(0)\} \cup \{(1)\} \cup \{(2)\} \cup \cdots \cup \{(n-1)\} \\ &= \{\} \cup \{0, 1, 2, \ldots, n-1\}.\\ &= \{0, 1, 2, \ldots, n-1\}.\\ \end{aligned}\]
A nice feature of this representation is that the number of elements of each set (its cardinality) is just the number itself. For example, the set representing 8 is \(\{0,1,2,3,4,5,6,7,\}\), which has 8 elements. Note these elements are generally sets containing other elements that are sets.
An interesting property of this construction is that \(a < b\) can be defined using set operations. Let \(a\) and \(b\) be two numbers and \(A\) and \(B\) be the two sets representing them respectively. Then \(a < b\) if and only if \(A \subset B\), where is \(A\) is a proper subset of \(B\).
Note the commas in the representation are extraneous, so juxtaposition is sufficient in expressing the ordinals. The number of \(\{\) symbols needed to represent \(n\) is \(2^n\). Similarly, \(2^n\) \(\}\) symbols are needed, resulting in a total of \(2^{n+1}\) required symbols. Thus 512 total symbols are needed to represent 8.
The artwork Braces of Eight uses just the symbols \(\{\) and \(\}\) to represent eight. The thickness of the symbols are varied to show the numeral 8. The background texture is made from randomly sized, colored, and placed 8's as shown below.
Related Works
Exhibition History
- Pi and Other Delights: The Mathematical Art of David A. Reimann, Composite Gallery, National Museum of Mathematics New York, New York, 20 May 2023 – TBD.
- Exhibition of Mathematical Art, Bridges Aalto 2022 Aalto University, Espoo, Finland, 1 August 2022 – 5 August 2022.
Publication History
- Conan Chadbourne, Robert Fathauer, Nathan Selikoff, and Bruce Torrence (Editors),
Bridges 2022 Art Exhibition Catalog, Tessellations Publishing, Phoenix, Arizona, pp. 62, 2022.
References
David A. Reimann,Artistic Depiction of Numbers Defined by Sets,in