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This new paragraph attempts to summarize the previous two paragraphs with a simple, clear, and rigorous statement of what can be claimed. |
→Some properties: Per WP:CALC. Is it possible to recognize it directly as a fractal or would mathematical sources be necessary there? |
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== Some properties ==
[[Image:Thue-Morse Squares.PNG|thumb|130px|right|Nested squares generated by successive iterations of Thue–Morse]]
[[File:ThueMorseQuadrant.gif|thumb|The pattern of points P<sub>n</sub> when x<sub>n</sub> and y<sub>n</sub> are the resulting horizontal and vertical binary sequences when Thue-Morse (starting with t<sub>1</sub>) is sorted into a quadrant shape {{citation needed}}]]
Because each new block in the Thue–Morse sequence is defined by forming the bitwise negation of the beginning, and this is repeated at the beginning of the next block, the Thue–Morse sequence is filled with ''squares'': consecutive strings that are repeated.
That is, there are many instances of ''XX'', where ''X'' is some string. Indeed, <math>X </math> is such a string if and only if <math>X =A,\, \overline{A},\, A \overline{A}A,</math> or <math>\overline{A}A \overline{A}</math> where <math>A=T_{k}</math> for some <math>k\ge 0</math> and <math>\overline{A} </math> denotes the bitwise negation of <math>A </math> (interchange 0s and 1s).<ref>{{cite journal|last1=Brlek|first1=Srećko|title=Enumeration of Factors in the Thue-Morse Word|journal=Discrete Applied Mathematics|year=1989| volume=24|pages=83–96|doi=10.1016/0166-218x(92)90274-e}}</ref> For instance, with <math>k=0</math>, we have <math>\ A= T_{0}=0 </math>, and the square <math>A \overline{A}AA \overline{A}A = 010010 </math> appears in <math>T </math> starting at the 16th bit. (Thus, squares in <math>T</math> have length either a power of 2 or 3 times a power of 2.)
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