This page has iteration diagrams.
A separate page explains Discrete Iteration, what these diagrams represent, and how they are formed.
LINK
On this page, I've captioned iteration diagrams that have flaws.
There are several kinds of glitches: twisted cycles, overlapping branches, crossing lines, etc.
I made up the terms as I went along. 'Floppers" and 'Flopdingers' are the same thing. LoL
Can you find the glitches in diagrams that I describe in words?
#25 is the first one to have a glitch, then #31, 33, 37, and so on.
Enjoy looking at them, discretely!
About the diagrams
These diagrams are 'Scalable Vector Graphics' (SVGs) — not images like PNG or JPG. Most browsers support the SVG format.
Read how the diagrams are generated here
LINK.
I apologize — none of the diagrams on this page have the nodes numbered. The other page has a few small graphs with labeled nodes. Here is 19 with labels...
19
Since the graphs on this page get rather large, labels aren't feasible.
My June 30, 2024 presentation has many labeled diagrams.
[3.5 MB PDF]
Observations
Look at the diagrams below...
7 is first n to have oscillator (2↔4)
11 is first to have a 4-cycle (3 → 9 → 4 → 5 → 3
17 is first to have a binary tree with 1 at the root, and 0 as a looper
19 is first to have a hexagon 6-cycle (4-16-9-5-6-17-4)
22 has TWO 4-cycles (4-16-14-20-4) (3,9,15,5,3)
23 has a 10-cycle (2-4-16-3-9-12-6-13-8-18-2)
and so on
Can you find the first 3-cycle?
Glitches in the graph layout.
As it is, the layout of some graphs are imperfect due to the limiting the amount of time the algorithm spends on the layout. The nodes have an initial layout (good problem in itself!) and then the algorithm repeatedly considers adjustments so as to minimize (maximize?) the "forces" between nodes in the graph.
Some nodes are initially placed in such a way that they get trapped in an equilibrium that is not the 'perfect' layout. Take a look at #23 — One of the nodes got started inside the cycle of nodes, and it is 'happy' where it is. Like wise, in #25, the 4-cycle is twisted, and has no cause to untwist itself.
It would be nice to generate a diagram into a file - in some format (xml, whatever), use an App to manipulate the arrangement, then update the file. For example, one could tweak node placement or sort the subgraphs into groups, and then have the layout algorithm do a 'cleanup' of your edited arrangement. In other words, provide some human assistance of the layout. Adobe Illustrator could be used to edit Postscript, but not do a layout cleanup. That's better than nothing, but I don't have access to Illustrator.
4 to 256
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23 -- One of the nodes "flopped" into inside of the cycle!
24
25 -- The 4-cycle (square) got twisted around.
26
27
28
29
30
31 -- Two 4-cycles (squares) got twisted around.32
33 -- One 4-cycle (square) got twisted.
34
35
36
37 -- two of the tiny trees feeding into the 6-cycle are swapped.
38 -- Twisted 6-cycle and some line crossings.
39
40
41 -- The 4-cycle (square) is twisted.
42 -- great stuff!
43
44 -- a 3-cycle is flattened. Dunno what that should look like!45
46 - Nice!
47
48
49
50 -- Twisted 4-cycle
51
52
53
54
55 -- Twisted 4-cycle
56
57 -- one line crossing.
58
59 -- Five nodes "flopped" — drawn on the inside of the large cycle.
60
61 -- two twisted 4-cycles
62 -- four of six 4-cycles are twisted!
63
64
65
66 -- just one leg of a 4-cycle is flopped over.
67
68
69
70
71
72
73
74 -- legs on two 6-cycles are crossing over
75 -- one 4-cycle is twisted.
76
77 -- one 4-cycle is twisted.
78
79
80
81
82
83 -- couple of floppers.
84
85 -- main subgraph is seriously messed up.
86
87 -- very minor layout of two lines feeding into a triangle.
88 -- twisted square
89 - Lovely!
90
91
92 -- minor flips. Same errors in the two 10-cycles.
93
94 - just two flopping errors.
95
96
97
98
99
100 -- one crunched 4 cycle.
101 - wow!
102 - minor symmetry violations of the two larger subgraphs.
103
104
105
106
107 -- bunch of floppers
108
109
110
111
112 -- symmetry problems in one subgraph.
113 - one triangle (3 cycle) is bent.
114 -- one 4-cycle twisted.
115 -- Cool! But, one line is On Top of another line. (4 O'Clock)
116
117
118 -- a few floppers
119
120
121
122
123
124
125 -- one glitch in lower right graph.
126
127
128 -- one node slightly out of place in lower left diagram.
129
130
131 -- 3 of the five 12-cycles have a single flopper.
132
133 -- OMG one hexagon is smashed.
134
135
136 -- minor layout error in lower left graph.
137
138
139 -- a couple floppers in the two larger cycles.
140
141 -- some floppers
142 -- some floppers, and two twisted 4-cycles
143 -- one twisted 4-cycle.
144
145
146 -- one twisted 6-cycle.
147
148 -- one badly twisted 6-cycle!
149 -- obvious
150 -- one twisted 4-cycle.
151 -- floppers and two twisted 4-cycles.
152 -- one minor minor flop on a 6-cycle.
153
154 -- two different kinds of 6-cycles are twisted.
155 -- unfortunate mess!
156
157 - clean!
158 - a couple floppers.
159 - minor flop errors
160
161 - minor flops
162 - floppers
163 -- whoa! Some floppers
164 -- one twisted 4-cycle
165 -- two twisted 4-cycles
166 - floppers
167 - floppers
168
169
170
171
172
173
174
175 -- one twisted 4-cycle
176 -- two twisted 4-cycles
177 - floppers
178
179
180
181 -- two twisted 4 cycles, legs crossed on the 6-cycle.
182
183 -- two different 4 cycles, twisted
184 - a couple floppers.
185 -- some unavoidable line crossings
186 -- four 4 cycles, (2 small, 2 larger), twisted
187 -- one bent 4-cycle, minor line crossings
188 -- a couple floppers in each of the cycles.
189
190 -- a twisted 6-cycle!
191 - floppers
192
193
194 -- one cross-over in lower right diagram
195
196
197 -- one little tree sucked into the middle.
198 -- one broken 4-cycle
199 -- a few floppers, not bad!
200
201 -- again, a few floppers, not bad!
202 -- Ha! The two 4-cycles are twisted.
203 -- One 6-cycle is twisted.
204
205 -- One 4-cycle is twisted. The other 4-cycle is amazing!
206
207 -- a few floppers.
208
209 -- a few floppers, and a twisted 4-cycle.
210
211 -- floppers in the large cycles; three twisted 4 cycles!
212 -- minor flopdingers.
213 -- floppers and a small twisted 4-cycle
214 -- flopdingers.
215
216 -- twisted 6-cycle.
217
218 -- two of the smaller 6-cycles have branches twisted.
219 -- small 6-cycle has branches swapped. Fancy 6-cycle has bigger branches swapped, deforming the hexagon.
220 -- one 4-cycle has a wild brnach.
221 -- smaller trees not attaching nicely.
222 -- one twisted 6-cycle; one pair of branches swapped in another 6-cycle..
223 -- a few flopdingers
224 -- all ok. some branches have uneven lengths.
225 -- just one 4-cycle twisted.
226 - So elegant!
227 -- just a few flopdingers
228
229 - Perfect!
230 -- one flopper in lower right, three in upper right.
231 -- a small 4-cycle got a little folded due to crowding?
232 -- tiny overlap glitch in upper left diagram.
233 -- obvious screw-up at 1 o'clock, 3 o'clock, and 9 o'clock.
234
235 -- flopdingers in separate graphs.
236 -- lotta floppers for some reason.
237 -- only one clean cycle without floppers
238 - fascinating!
239 -- Ha! an Octagon twisted into a cross-like loop.
240
241 -- a twisted 4-cycle and one branch overlaying another.
242 -- flopdingers in the larger cycles, and two twisted 4-cycles.
243 -- Sweet! Just the usual floppers.
244 -- Oh oh.. Three of the 4-cycles got twisted.
245 - so interesting!
246 -- just one twisted 4-cycle.
247 -- just one twisted 6-cycle (hexagon).
248 -- two twisted 4-cycles.
249 -- some floppers
250 -- one small twisted 4-cycle, otherwise gorgeous!
251 -- flopdingers and a trivial twisted 4-cycle.
252
253 -- a few floppers on the two large cycles.
254 -- one twisted hexagon
255-- strange complication in the upper right diagram. One 4-branched tree could join the other one on the right side, and adjustments made.
256
Index of Iteration Diagrams
ATTENTION: The pages below do not seem to load on my oldish iPhone. The pages load OK on my iPad and Mac Mini. Each page has 256 Scalable Vector Graphic (SVG) images.
