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# 4. Some interesting graphs produced by the set of L states

In this section we study 3D and 2D graphs produced by L states of chocolate games that satisfy inequalities for .

Example 4.1   Here we study graphs made by the set of L states for the chocolate game of Definition 3.1 which satisfies the inequality .
We denote by the set of L state of this chocolate.

Fig 4.1 is a 3D graph of .

By we define a sequence of sets , . By plotting for we get Fig 4.2.
We use the following Mathematica program. Note that we join points with curved segments by Mathematica command " ".

LS[1]=ppo[50];Clear[d]; Do[ d[1,k] = Map[{#[[2]], #[[3]]} &,    Select[LS[1], #[[1]] == k &]], {k, 0, 50}];ListPlot[Table[d[1,k], {k, 1, 50}], Joined -> True]

By plotting for we get Fig 4.3.

Example 4.2   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality .
We denote by the set of L state of this chocolate.

Fig 4.4 is a 3D graph of .

By we define a sequence of sets for . By plotting for we get Fig 4.5.

By plotting for we get Fig 4.6.

Example 4.3   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality .
We denote by the set of L state of this chocolate.

Fig 4.7 is a 3D graph of .

By we define a sequence of sets . By plotting for we get Fig 4.8.

By plotting for we get Fig 4.9.

Example 4.4   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality .
We denote by the set of L state of this chocolate.

Fig 4.10 is a 3D graph of .

By we define a sequence of sets for . By plotting for we get Fig 4.11.

By plotting for we get Fig 4.12.

Example 4.5   Here we study graphs made by the set of L states for the chocolate game that satisfies the inequality .
We denote by the set of L state of this chocolate.

Fig 4.13 is a 3D graph of .

By we define a sequence of sets for . By plotting for we get Fig 4.14.

By plotting for we get Fig 4.15.

For the 3D graphs are beautiful, but 2D graphs are not beautiful, however, for the 3D graphs and 2D graphs are both beautiful.

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