User:Jon Awbrey/GRAPHICS

< User:Jon Awbrey
Revision as of 18:52, 7 January 2011 by Jon Awbrey (talk | contribs) (+ Miscellaneous Examples (Reduction 8:1))
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Animations

Riffs 1 to 60

 

Rotes 1 to 60

 

Image Gallery

Reduction 6:1

Cacti

Image Scale
  117 px

20 px
  117 px

20 px

Riffs

Integer Riff Scale
1    
2   117 px

20 px
3   240 px

40 px
4   233 px

40 px
5   384 px

65 px
6   384 px

65 px
7   377 px

65 px
8   384 px

65 px
9   240 px

40 px
10   528 px

90 px
11   528 px

90 px
12   384 px

65 px
13   384 px

65 px
14   521 px

90 px
15   528 px

90 px
16   377 px

65 px
17   521 px

90 px
18   384 px

65 px
19   528 px

90 px
20   528 px

90 px
21   521 px

90 px
22   672 px

115 px
23   384 px

65 px
24   672 px

115 px
25   384 px

65 px
26   528 px

90 px
27   384 px

65 px
28   521 px

90 px
29   528 px

90 px
30   672 px

115 px
31   672 px

115 px
32   528 px

90 px
33   672 px

115 px
34   665 px

115 px
35   521 px

90 px
36   384 px

65 px
37   384 px

65 px
38   672 px

115 px
39   672 px

115 px
40   816 px

135 px
41   528 px

90 px
42   665 px

115 px
43   521 px

90 px
44   672 px

115 px
45   528 px

90 px
46   528 px

90 px
47   528 px

90 px
48   384 px

65 px
49   377 px

65 px
50   528 px

90 px
51   665 px

115 px
52   528 px

90 px
53   521 px

90 px
54   528 px

90 px
55   672 px

115 px
56   809 px

135 px
57   672 px

115 px
58   672 px

115 px
59   665 px

115 px
60   672 px

115 px
64   384 px

65 px
81   377 px

65 px
105   665 px

115 px
128   521 px

90 px
165   816 px

135 px
256   528 px

90 px
512   384 px

65 px
2010   1104 px

185 px
2500   528 px

90 px
65536   521 px

90 px
802701   1248 px

210 px
123456789   1296 px

220 px

Rotes

Integer Rote Scale
1   117 px

20 px
2   233 px

40 px
3   233 px

40 px
4   377 px

65 px
5   233 px

40 px
6   473 px

80 px
7   377 px

65 px
8   377 px

65 px
9   473 px

80 px
10   473 px

80 px
11   233 px

40 px
12   617 px

105 px
13   473 px

80 px
14   617 px

105 px
15   473 px

80 px
16   521 px

90 px
17   377 px

65 px
18   713 px

120 px
19   377 px

65 px
20   617 px

105 px
21   617 px

105 px
22   473 px

80 px
23   473 px

80 px
24   617 px

105 px
25   473 px

80 px
26   713 px

120 px
27   473 px

80 px
28   761 px

130 px
29   473 px

80 px
30   713 px

120 px
31   233 px

40 px
32   377 px

65 px
33   473 px

80 px
34   617 px

105 px
35   617 px

105 px
36   857 px

145 px
37   617 px

105 px
38   617 px

105 px
39   713 px

120 px
40   617 px

105 px
41   473 px

80 px
42   857 px

145 px
43   617 px

105 px
44   617 px

105 px
45   713 px

120 px
46   713 px

120 px
47   473 px

80 px
48   617 px

105 px
49   473 px

80 px
50   713 px

120 px
51   617 px

105 px
52   857 px

145 px
53   521 px

90 px
54   713 px

120 px
55   473 px

80 px
56   761 px

130 px
57   617 px

105 px
58   713 px

120 px
59   377 px

65 px
60   905 px

155 px
64   617 px

105 px
81   617 px

105 px
105   857 px

145 px
128   521 px

90 px
165   713 px

120 px
256   521 px

90 px
512   617 px

105 px
2010   1145 px

190 px
2500   1001 px

170 px
65536   665 px

115 px
802701   1961 px

330 px
123456789   2048 px

345 px

Reduction 10:1

Cacti

Image Scale
  117 px

12 px
  117 px

12 px

Riffs

Integer Riff Scale
1    
2   117 px

12 px
3   240 px

24 px
4   233 px

24 px
5   384 px

38 px
6   384 px

38 px
7   377 px

38 px
8   384 px

38 px
9   240 px

24 px
10   528 px

54 px
11   528 px

54 px
12   384 px

38 px
13   384 px

38 px
14   521 px

54 px
15   528 px

54 px
16   377 px

38 px
17   521 px

54 px
18   384 px

38 px
19   528 px

54 px
20   528 px

54 px
21   521 px

54 px
22   672 px

68 px
23   384 px

38 px
24   672 px

68 px
25   384 px

38 px
26   528 px

54 px
27   384 px

38 px
28   521 px

54 px
29   528 px

54 px
30   672 px

68 px
31   672 px

68 px
32   528 px

54 px
33   672 px

68 px
34   665 px

68 px
35   521 px

54 px
36   384 px

38 px
37   384 px

38 px
38   672 px

68 px
39   672 px

68 px
40   816 px

82 px
41   528 px

54 px
42   665 px

68 px
43   521 px

54 px
44   672 px

68 px
45   528 px

54 px
46   528 px

54 px
47   528 px

54 px
48   384 px

38 px
49   377 px

38 px
50   528 px

54 px
51   665 px

68 px
52   528 px

54 px
53   521 px

54 px
54   528 px

54 px
55   672 px

68 px
56   809 px

82 px
57   672 px

68 px
58   672 px

68 px
59   665 px

68 px
60   672 px

68 px
64   384 px

38 px
81   377 px

38 px
105   665 px

68 px
128   521 px

54 px
165   816 px

82 px
256   528 px

54 px
512   384 px

38 px
2010   1104 px

110 px
2500   528 px

54 px
65536   521 px

54 px
802701   1248 px

125 px
123456789   1296 px

130 px

Rotes

Integer Rote Scale
1   117 px

12 px
2   233 px

24 px
3   233 px

24 px
4   377 px

38 px
5   233 px

24 px
6   473 px

48 px
7   377 px

38 px
8   377 px

38 px
9   473 px

48 px
10   473 px

48 px
11   233 px

24 px
12   617 px

62 px
13   473 px

48 px
14   617 px

62 px
15   473 px

48 px
16   521 px

54 px
17   377 px

38 px
18   713 px

72 px
19   377 px

38 px
20   617 px

62 px
21   617 px

62 px
22   473 px

48 px
23   473 px

48 px
24   617 px

62 px
25   473 px

48 px
26   713 px

72 px
27   473 px

48 px
28   761 px

76 px
29   473 px

48 px
30   713 px

72 px
31   233 px

24 px
32   377 px

38 px
33   473 px

48 px
34   617 px

62 px
35   617 px

62 px
36   857 px

86 px
37   617 px

62 px
38   617 px

62 px
39   713 px

72 px
40   617 px

62 px
41   473 px

48 px
42   857 px

86 px
43   617 px

62 px
44   617 px

62 px
45   713 px

72 px
46   713 px

72 px
47   473 px

48 px
48   617 px

62 px
49   473 px

48 px
50   713 px

72 px
51   617 px

62 px
52   857 px

86 px
53   521 px

54 px
54   713 px

72 px
55   473 px

48 px
56   761 px

76 px
57   617 px

62 px
58   713 px

72 px
59   377 px

38 px
60   905 px

90 px
64   617 px

62 px
81   617 px

62 px
105   857 px

86 px
128   521 px

54 px
165   713 px

72 px
256   521 px

54 px
512   617 px

62 px
2010   1145 px

115 px
2500   1001 px

100 px
65536   665 px

68 px
802701   1961 px

196 px
123456789   2048 px

205 px

Riffs in Numerical Order

\(\text{Riffs in Numerical Order}\!\)

 


\(1\!\)


\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

 


\(\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

 


\(\text{p}_\text{p}\!\)


\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

 


\(\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

 


\(\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

 


\(\text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

 


\(\text{p}^{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

 


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

 


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

 


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

 


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

 


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

 


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

 


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

 


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

 


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

 


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

 


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

 


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Rotes in Numerical Order

 


\(1\!\)


\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

 


\(\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

 


\(\text{p}_\text{p}\!\)


\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

 


\(\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

 


\(\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

 


\(\text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

 


\(\text{p}^{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

 


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

 


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

 


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

 


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

 


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

 


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

 


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

 


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

 


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

 


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

 


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

 


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

 


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

 


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

 


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

 


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

 


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

 


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

 


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

 


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Miscellaneous Examples (Reduction 8:1)

\(\text{Integers, Riffs, Rotes}\!\)
\(\text{Integer}\!\) \(\text{Riff}\!\) \(\text{Rote}\!\)
\(1\!\)    
\(2\!\)    
\(3\!\)    
\(4\!\)    
\(2010\!\)    
\(2011\!\)    
\(2500\!\)    
\(802701\!\)    
\(123456789\!\)    

Cactus Graphs

Hi Res


  117 px → 20 px
  117 px → 20 px
  117 px → 20 px
  117 px → 20 px
  290 px → 50 px
  386 px → 65 px
  204 px → 35 px
  348 px → 60 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  386 px → 65 px
  530 px → 90 px
  530 px → 90 px


Lo Res


 
 
 
 


Differential Logic

ASCII Graphics

Series 1

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /                 o                 \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |                 |%%%%%|                 |   |
|   |        P        |%%%%%|        Q        |   |
|   |                 |%%%%%|                 |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
Figure 22-a.  Conjunction pq : X -> B
o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o.->-.o                 o   |
|   |    p(q)(dp)dq   |%\%/%|  (p)q dp(dq)    |   |
|   | o---------------|->o<-|---------------o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                  (p)(q) dp dq                   |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Ef =              p  q   (dp)(dq)               |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-b.  Enlargement E[pq] : EX -> B
o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |       (dp)dq    |%%%%%|    dp(dq)       |   |
|   | o<--------------|->o<-|-------------->o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                      dp dq                      |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Df =              p  q  ((dp)(dq))              |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-c.  Difference D[pq] : EX -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o                                    |
|           /                     \                                   |
|          /                       \                                  |
|         /                         \                                 |
|        /                           \                                |
|       /                             \                               |
|      /                               \                              |
|     /                                 \                             |
|    o                                   o                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                 G                 |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    o                                   o                            |
|     \                                 /                             |
|      \                               /                              |
|       \                           T /                               |
|        \             o<------------/-------------o                  |
|         \                         /                                 |
|          \                       /                                  |
|           \                     /                                   |
|            o-------------------o                                    |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 23.  Elements of a Cybernetic System

Series 2

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       /%\                       \         |
|        /                       /%%%\                       \        |
|       /                       /%%%%%\                       \       |
|      /                       /%%%%%%%\                       \      |
|     /                       /%%%%%%%%%\                       \     |
|    o                       o%%%%%%%%%%%o                       o    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |          P            |%%%%%%%%%%%|            Q          |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    o                       o%%%%%%%%%%%o                       o    |
|     \                       \%%%%%%%%%/                       /     |
|      \                       \%%%%%%%/                       /      |
|       \                       \%%%%%/                       /       |
|        \                       \%%%/                       /        |
|         \                       \%/                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-1.  Proposition pq : X -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o<-----------------o----------------->o          |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-2.  Tacit Extension !e![pq] : EX -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o----------------->o<-----------------o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-1.  Enlargement E[pq] : EX -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |             (dp) dq   |           |   dp (dq)             |    |
|    |          o<---------------->o<---------------->o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-2.  Difference Map D[pq] : EX -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   o   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |                   (dp)/ dq     dp \(dq)                   |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 v     \  dp dq  /     v                 /     |
|      \               o<--------------------->o               /      |
|       \                       \     /                       /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-1.  Differential or Tangent d[pq] : EX -> B
o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   dp dq   |                       |    |
|    |            o<------------------------------->o            |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |     o     |                       |    |
|    o                       o     ^     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    dp | dq                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-2.  Remainder r[pq] : EX -> B

JPEG Graphics

Series 1

 
\(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\)
 
\(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

 
\(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

Series 2

 
\(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\)
 
\(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

 
\(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

 
\(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

 
\(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\)

 
\(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\)

Propositional Equation Reasoning Systems

Analysis of contingent propositions

  (26)
  (27)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}\)
  (28)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\)
  (29)

Equation 1 : Proof 1

  (30)

Equation 1 : Proof 2

Single Image Version
  (31)
Serial Image Version
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(31)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(33)

Equation 1 : Proof 3

  • Variant 1
  1. start
  2. cast p
  3. dom
  4. can
  5. empty
  6. can
  7. cast q
  8. dom
  9. can
  10. dom
  11. spike
  12. can
  13. cast r
  14. can
  15. empty
  16. spike
  17. can
  • Variant 2
  1. start
  2. cast p
  3. dom
  4. can
  5. empty
  6. can
  7. cast q
  8. can
  9. dom
  10. can
  11. spike
  12. can
  13. cast r
  14. can
  15. empty
  16. spike
  17. can
Variant 1
o-----------------------------------------------------------o
| Equation E_1.  Proof 3.                                   |
o-----------------------------------------------------------o
|                                                        1  |
|               q o   o r   q o r                           |
|                 |   |       |                             |
|               p o   o p   p o                             |
|                  \ /        |                             |
|                   o---------o                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        o                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        @                                  |
|                                                           |
o==================================< CAST "p" >=============o
|                                                        2  |
|          q   r    q r    q   r    qr                      |
|          o   o     o     o o o o   o o                    |
|          |   |     |     |/  |/    |/                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        3  |
|          q   r    q r                                     |
|          o   o     o       o   o     o                    |
|          |   |     |      /   /     /                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        4  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness >============o
|                                                        5  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o           o                          |
|             \     /            |                          |
|              \   /             |                          |
|               \ /              |                          |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        6  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|                o                                          |
|                |                                          |
|                |                                          |
|                |                                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "q" >=============o
|                                                        7  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o r                        |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        8  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o                          |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        9  |
|          r     r         r                                |
|          o     o         o                                |
|          |     |         |                                |
|          o     o     o   o     o                          |
|         /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                       10  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o     o         o                          |
|         /      |      \        |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Spike >================o
|                                                       11  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o           o                              |
|         \     /            |                              |
|          \   /             |                              |
|           \ /              |                              |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       12  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o                                          |
|         \     /                                           |
|          \   /                                            |
|           \ /                                             |
|            o                                              |
|            |                                              |
|            |                                              |
|            |                                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "r" >=============o
|                                                       13  |
|                      o     o                              |
|                      |     |                              |
|      o     o         o     o                              |
|      |     |         |     |                              |
|      o     o         o     o                              |
|     /      |        /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       14  |
|                      o     o                              |
|                     /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness & Spike >====o
|                                                       15  |
|        o               o                              16  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       17  |
|          r o-------o---o r                                |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< QED >==================o
(40)
Variant 2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(40)

Praeclarum Theorema : Proof by CAST

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(23)