ABSTRACT
(Index to the section of ESA introducing or adjusting the term is provided in parentheses.)
B is the number of possible subdivisions available in the image. B = nS /VCF. (Section 4.2.4)
C is the number of segments in a cycle. C = S /GCD( S , P ). C is in cell M9 in the equations column of the Excel file. (Section 5.1)
CPF stands for Centered-Point Flowers. This is the simplest variation on the string art model discussed in Part III. After each vertex jump, there is a jump to the center and a jump back out to the vertex. (Chapter 15)
Circuit: A circuit is complete once the starting point for the image (in ESA this is always the top of the circle (except in the non-polygonal Chapter 19)) is achieved as an endpoint. (Section 2.2.1)
Continuously-drawn: An image is continuously-drawn if line segments are connected from one to another following a rule until the initial starting point is obtained as the end point of a segment. This applies both to polygons and stars (Section 2.2.1) and images with subdivisions (Section 4.2.1).
Cycle: A partial image consisting of the line segments needed to get from Level 0 back to Level 0. Level 0 points are vertices of the n -gon. This is a non-mathematical usage of this term inspired by a spirograph. In mathematical terms, this would more aptly be described as a cycle generator and the completed image is a cycle. (Section 5.1)
Curved-tip stars: A curved-tip star is created when the vertex frame is a star and P < S . (11.2) Such images include the vertex frame and are useful when initially working with jump-sets in Part III, especially in Chapter 17.
DL stands for Drawn Lines in the web model. (Used throughout ESA but discussed formally in Section 25.4.)
DP stands for Drawing Progress in the web model. (Used throughout ESA but discussed formally in Section 25.4.)
DS stands for Drawing Speed in the web model. (Used throughout ESA but discussed formally in Section 25.4.)
Degree of rotational symmetry: The degree of rotational symmetry is the number of different vertices it can be rotated to, for which it matches the original. (Section 6.4.1)
Distinct images: Images based on different n , S , P , J values may look identical to one another—they may produce the same static image. There are a variety of reasons this may happen. Distinct images have different n , S , P , J values AND look different from one another. (Section 4.3) See also static image. (Section 6.2)
Donut hole: The donut hole is the area inside the innermost concentric circle of subdivisions. (Section 10.2.1)
Drawing Mode is the name of the drop-down menu in the web model. Four drawing choices are available: FCLD, FCFLD, SLD, and SLOD. (Section 25.4)
E is the number of the parent polygon vertex that occurs at the end of the first cycle. E is in cell M14 in the equations column of the Excel file. (Section 5.2)
ESA stands for Electronic String Art. (Section 1.1.1)
Explainer: Sections within this book are, in general, very short and targeted. They have been created to explain something, and hence they are sometimes called explainers instead of sections. Care has been taken to provide specific numerical reference to explainers outside the “just the basics” explainers noted in Section 1.2. (Section 1.2)
F is the Level of the first endpoint in the image. F is in cell M28 in the equations column of the Excel file. (Section 7.2)
FCFLD stands for Fixed Count Fading Line Drawing Mode in the web model. Use if you have large numbers of lines in the image. This drawing mode fills in the image DL lines at a time. (Section 25.4)
FCLD stands for Fixed Count Line Drawing Mode in the web model. This is typically the most heavily used line drawing mode. (Section 25.4)
GCD: The greatest common divisor of two numbers is the largest factor common to both. (First mentioned in Section 2.2.2 but discussed at length in Chapter 21)
Image: Term used for a completed graph. (Section 1.1.1)
Image density: Density relates to the portion of potential subdivisions that are used in the image. (Section 5.4)
J is the number of J umps between vertices. When J = 1, the resulting image is a polygon. If J > 1, stars can emerge. (Sections 1.1 and 2.2.1)
Jump Set: A jump set is a pattern of jumps that is repeated. (Section 16.1)
Just-over and just-under multiples: Interesting images often occur when one parameter is close to but not quite a multiple of another. This is seen in numerous places but notably in Section 2.5, stars as rotating polygons ( n = m · J ± a where m is a whole number and a is a small whole number). In Part II, this is seen in Section 7.2, One Level Change images, Section 8.1 Shape-Shifting polygons, as well as with numerous images in Chapters 11–13. A more mathematical approach to this is in Chapter 24 on modular multiplicative inverses and negative MMIs.
k is used to define a range of values of a parameter. This example is from Section 2.3.3. Suppose you want to list the values of n ≥ 4 that are multiples of 4. You could do this by stating n = 4, 8, … or by saying n = 4 k for k ≥ 1. By contrast, if you want n larger than 5 that are divisible by 2 but not 4, you would say n = 4 k +2 for k ≥ 1. The letter k is also used to show the first k lines (using the toggle switch in cell B10 value in C11) in multiple string art Excel files from Part II starting with 3.0.3 (but in Chapter 10, r describes this attribute as noted at r below).
L is the number of L ine segments in the image. An image may appear to have fewer line segments than listed for a couple of reasons: 1) segments may overlap; and 2) segments may be part of the same line. An example of the first is the vertical line that results whenever n is even and J = n /2. A simple example of the second is when ( n , S , P , J ) = (3, 2, 1, 1). The resulting triangle has L = 6, 2 on each of the three sides. (Section 4.1)
L is also used to denote the number of lines in a sub-image, for example the lines in a single-step. Single-step images require VCF = SCF = 1. The sub-image is single-step when L satisfies ±1 = L·P MOD n·S . (Section 8.5.1, see also Section 11.6.1)
Levels: There are INTEGER( S /2) subdivision-point-created concentric interior circles. (INTEGER is the integer portion of a fraction so INTEGER(5/2) = INTEGER(2.5) = 2 (not 3 as would be the case were we to round up to the next nearest integer as is common mathematical practice).) Points at Level 0 are polygonal vertices, and Level INTEGER( S /2) is the smallest circle (this is the boundary of the donut hole discussed in Section 10.2.1). (Section 7.1)
M is the number of cycles in an image. M must be a factor of n . M is in cell M11 in the equations column of the Excel file. (Section 5.1)
MA stands for M athematical A pproach. It signifies that the explainer in question [or the bracketed part of an explainer] may not be accessible to primary school audiences.
MBS stands for Modified Brunes Star. (Sections 10.2.3, 14.8, and 14.9)
MMI stands for Modular Multiplicative Inverse. Two numbers are multiplicative inverses if their product is 1. If one number is a whole number a , the other is a fraction 1/ a . In modular arithmetic, two numbers a and c are modular multiplicative inverses mod b > 0 if a·c is 1 more than a multiple of b . This is the mathematical description of “just-over multiples.” (Examined in Chapter 24 but used extensively throughout the book (sometimes without attribution as such, for example in Section 8.4).)
MOD is the mathematical way of saying remainder upon division by a number. For example, 5 = 1 MOD 2 since 5 = 2·2+1 or 17 = 2 MOD 5 since 17 = 3·5+2 but so is 2, 7, 12, and so forth, MOD 5. (Chapter 23)
Mystic Rose: A mystic rose is an image in which each vertex is connected to every other vertex. The number of distinct lines in an n -point mystic rose is n ·( n -1)/2. (Introduced in Section 18.1 but discussed more formally in Section 18.2.)
n -gon: An n -sided polygon. (Sections 1.1.1 and 2.1)
n -gram: An n -sided star. This is a generalization of a pentagram. (Section 1.1.1 and 2.2)
n,J -star: An explicit version of an n -gram. Note that an n,J -star only has n points if VCF = 1. (Section 2.2)
nMMI is negative MMI. In modular arithmetic, two numbers a and c are nMMI MOD b > 0 if a·c is 1 less than a multiple of b . This is the mathematical description of “just-under multiples.” (Examined in Chapter 24 but used extensively throughout the book; see, for example, Section 2.5.1 without attribution or Section 13.1 with attribution.)
NP stands for the Non-Polygon model (with user-controlled vertices) presented in Chapter 19.
One-level change images move in and out one level at a time. (Section 7.2)
One-time-around images: If T = 1 the image is called a one-time-around image. (Section 5.2)
P is the number of subdivisions between P oints. P is a positive whole number that provides the counting rule for producing images. The image is created by connecting subdivision endpoints that are P subdivisions apart from one another with a line segment starting at the top and ending when the last endpoint completes the circuit by being at the starting point. (Sections 1.1.1 and 3.2)
Polygon: A polygon occurs if the line segments comprising the image do not cross over one another except at the common endpoint. A polygon is regular if all vertices are equally spaced around a circle. Also called an n -gon. (Section 2.1)
Polygram: Another name for a star, for example, a pentagram is a 5,2-star. Also called an n -gram.
Porcupine: An almost halfway-around image. (Sections 11.3 and 11.4 but introduced informally in Section 1.1.1)
r is the remainder when one number is divided by another. This concept is used informally throughout this book but is formally defined in Section 23.1. See also Section 7.3. In the Chapter 10 Excel files, r is used in place of k (the show first r lines toggle in cell B10 value in C11) because k is used to describe functional forms in that chapter.
S is the number of equally spaced S ubdivisions between successive vertices in the vertex frame. S is a positive whole number. (Sections 1.1.1 and 3.1)
SCF: The subdivision common factor, SCF, is given by SCF = GCD( P , S * v used ). (Sections 4.1, 15.2, and 16.1)
SLD stands for Single Line Drawing Mode in the web model. This mode shows DL lines from the image regardless of DP. (explained in Section 25.4 but heavily used in Sections 8.4, 11.9, and 12.8)
SLOD stands for Single Line Overlaid Drawing Mode in the web model. This mode overlays DL lines on a completed image. (Explained in Section 25.4 but heavily used in Sections 9.4, 12.5, and 12.10)
Shape-Shifting Polygons involve sub-images that change shape over the course of a cycle. Chapter 8 is devoted to a general discussion of this topic, but the most interesting version may well be 3SST noted next.
3SST is short-hand for Three Shape-Shifting Triangles, examined in Sections 8.4 and 8.6. Variations on 3SST are seen in Section 9.5, which compares the concepts of single-step and smallest-step, and in Section 12.8, which provides a version of 3SST using quivering triangles. Section 8.6 notes a similar image that is 5SST with DL = 11.
single-step images: Single-step images occur when a subset of lines in the image ends at 1 more or 1 less than the top of the image. The image will then appear to fill in sequentially around that subset of lines. (Section 8.5.1)
smallest-step images: When SCF > 1, one cannot have single-step images, but one can obtain sub-images that are as close to the top as possible. That is what the smallest-step means. (Sections 9.4 and 9.5)
Spinning Needle Stars: This is an image that is the smallest-step of length 2 that is one-time-around. Formally discussed in Section 11.8.1, but it is also examined elsewhere, most notably in Sections 11.8.2, 9.6, and 11.12.
Star: A star occurs when the image has segments that cross over other segments at points other than their endpoints. Also called an n -gram. Often a star is described as a n , J -star where n and J are the number of vertices and the number of jumps, so a 5,2-star is a common pentagram. (Sections 1.1 and 2.2)
Static image: The notion of the same static image typically refers to the idea that if you replace J with n - J you have the same image, but it is drawn in the reverse direction. (Section 6.2) See also Distinct image (as a function of P ). (Section 4.3)
Sub-image: This is a partial image used to explain a specific point or understand the overall structure of the image. The most common sub-images are cycles (Chapter 5) or the lines required for a single-step image (Section 8.5.1) or a smallest-step image (Section 9.4).
T is the number of times around vertices of the parent polygon are added to create the image. This is most easily seen using the FCLD Drawing Mode on the companion website. The easiest to see are one-time-around and two-time-around images. T is in cell M25 in the equations column of the Excel file. (Section 5.2)
V is the number of user-determined vertices that are used in the non-polygonal model introduced in Chapter 19. It takes the place of n and J in that chapter. (Section 19.1)
VF: The Vertex Frame, VF, is the set of line segments that include all possible subdivisions. The vertex frame is solely determined by n and J . Visualize the vertex frame by setting S = P . (Sections 1.1 and 4.2.2)
VCF: The vertex common factor, VCF, is given by VCF = GCD( J , n ). (Sections 2.2.2, 4.1, and 16.1)
v used : The number of polygon vertices used is given by v used = n /VCF. (Sections 4.1 and 16.1) Note that a vertex being used does not mean that it is part of the final image. (Section 5.4.2. But see Section 11.8.1 for very clear examples of this notion.)
(Punctuation guide. Individual letters are italicized and bold, acronyms are normal typeface unless necessary for emphasis. MA is always bold not italicized. Continuously-drawn uses a dash to emphasize the connectedness of this idea.)
