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thankyou for making this page, whoever did. The trouble I am having with it is those unaware of degree level maths facts and jargon have to spend ages looking up other pages. I often have this problem with maths pages of
Knowledge. Can someone please help, because fractals are pretty and I would like
360:
And by the way, as is well known, any number whose binary expansion ends in a sequence of all 0's can be re-expressed in a binary expression ending in a sequence of all 1's, and vice versa. SO: Do both sequencs of binary coefficients give the identical p_x ? Whether the answer is Yes or No, this
3107:
But this is the same thing. de Rham himself gives both of these constructions in his original paper (the 1957 paper cited in the article). He spends some time computing the length of the cut-polygonal-arc construction, as that is the construction where it is easy to compute the length.
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so one has to fiddle about to re-obtain the above relationships. Note that ALL of these curves have the period-doubling symmetry, NOT just the question mark. That is, there are appropriate variants of S and T that describe ALL of these curves!
630:
2672:, so the points belonging to the curve verify that (S(x),y/2) and (T(x),(1+y)/2) are also on the curve. Then the following mappings generate the 2-dimensional curve corresponding to the graph of the Minkowski's question mark function:
632:, maybe saying a word about this construction, because the composition of an infinite sequence of functions is not something natural (and the subject is not even mentioned in the linked page about function composition),
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expression most often design "curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio
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converges uniformly (because of contraction mappings) toward a continuous curve (because of uniform convergence), that we will define to be the De Rham curve.
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denotes function composition. It can be shown that . The collection of points p_x, parameterized by a single real parameter x, is known as the de Rham curve.
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2062:{\displaystyle (S^{a_{1}}\circ R\circ S^{a_{2}}\circ R\circ \cdots S^{a_{n}})(1)=(S^{a_{1}}\circ T^{a_{2}}\circ S^{a_{3}}\circ \cdots )(1)}
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Only then should the number x be defined and described as a way of parametrizing the collection of points, which can now be called {p_x}.
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2901:{\displaystyle d_{1}\left({\begin{matrix}x\\y\end{matrix}}\right)=\left({\begin{matrix}1/(2-x)\\(1+y)/2\end{matrix}}\right)}
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2777:{\displaystyle d_{0}\left({\begin{matrix}x\\y\end{matrix}}\right)=\left({\begin{matrix}x/(x+1)\\y/2\end{matrix}}\right)}
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Instead, the article should read something like "Given a sequence {b_k}, k = 1,2,3,... of 0's and 1's, ... etc.
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I think the definition of the maps that generate the
Minkowski's question mark function are wrong.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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2254:, but this is not what is requested from a De Rham curve: we need to have contraction maps from
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referring to the sequence b_1, b_2, . . . of 0's and 1's is putting the cart before the horse.
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page should be dedicated to that curve, and the content of the current page should be moved to
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625:{\displaystyle c_{(b_{k})}=d_{b_{1}}\circ d_{b_{2}}\circ \cdots \circ d_{b_{k}}\circ \cdots }
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But this would be long and technical. A more intuitive solution could be to:
1566:, k=0,1,…,2^n, in this order (This would benefit from a nice illustration!).
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Show that this map is continuous (it can stay in the properties section).
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Here, each b_k is understood to be an integer, 0 or 1. Consider the map
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Additionally, I think this article lack of historical background.
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I think you're right about the order, to be rigorous we should:
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Let x be a real number in the interval , having binary expansion
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Show how the construction relates to iterated function systems.
15:
1813:, it is true that every rational whose continued fraction is
798:{\displaystyle x_{(b_{k})}=\sum _{k=1}^{\infty }b_{k}2^{-k}}
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http://www.insa-rennes.fr/~jmerrien/articles/drla3.pdf
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2291:{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}
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1806:{\displaystyle T(x)=(R\circ S\circ R)(x)=1/(2-x)}
1467:{\displaystyle p_{1/2}=d_{0}(p_{1})=d_{1}(p_{0})}
1622:A mistake in Minkowski's question mark function?
1189:define recursively for each dyadic rational (
8:
3025:{\displaystyle \omega :(1-2\omega ):\omega }
2321:{\displaystyle \mathbb {R} \to \mathbb {R} }
101:, which collaborates on articles related to
286:to understand them. Edit this page please.
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1374:{\displaystyle p_{q}=d_{1}(p_{2q-1})}
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218:This article is within the scope of
157:This article is within the field of
95:This article is within the scope of
1309:{\displaystyle p_{q}=d_{0}(p_{2q})}
323:x = Sum from 1 to ∞ of b_k * 2^(-k)
38:It is of interest to the following
3071:{\displaystyle \omega \in (0,1/2)}
2598:{\displaystyle ?(T(x))=(1+?(x))/2}
767:
635:show that the contraction mapping
520:introduce the contraction mapping
347:defining the number x rather than
14:
3158:Low-priority mathematics articles
3078:is a given parameter" (see, e.g.
1524:formed by joining all the points
238:Knowledge:WikiProject Mathematics
3153:Start-Class mathematics articles
3143:Systems articles in chaos theory
241:Template:WikiProject Mathematics
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3138:Mid-importance Systems articles
2525:{\displaystyle ?(R(x))=R(?(x))}
2409:The relations that we need are
258:This article has been rated as
135:This article has been rated as
3118:04:10, 21 September 2020 (UTC)
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2372:{\displaystyle c_{x}(1)=?(x)}
2298:, when S and T are maps from
2247:{\displaystyle c_{x}(1)=?(x)}
2167:{\displaystyle d_{1}(x)=T(x)}
2116:{\displaystyle d_{0}(x)=S(x)}
1616:00:35, 11 November 2008 (UTC)
315:This introductory paragraph:
232:and see a list of open tasks.
115:Knowledge:WikiProject Systems
3148:WikiProject Systems articles
3133:Start-Class Systems articles
2532:from which we conclude that
1677:{\displaystyle S(x)=x/(x+1)}
361:issue needs to be addressed.
118:Template:WikiProject Systems
1559:{\displaystyle p_{k/2^{n}}}
1497:, consider the broken line
960:, which defines a map from
953:{\displaystyle p_{(b_{k})}}
913:{\displaystyle x_{(b_{k})}}
708:{\displaystyle p_{(b_{k})}}
668:{\displaystyle c_{(b_{k})}}
371:02:32, 23 August 2008 (UTC)
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2605:. In the plane, the point
675:has an unique fixed point
141:project's importance scale
3088:`De Rham curve (Fractal)´
1223:{\displaystyle q=k/2^{n}}
305:) 11:24, 8 December 2007
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2637:belongs to the curve if
1718:{\displaystyle R(x)=1-x}
1114:introduce the condition
808:Introduce the condition
264:project's priority scale
2980:In the literature, the
2379:is not the fixed point
1057:and their fixed points
490:introduce the sequence
433:and their fixed points
221:WikiProject Mathematics
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311:Needs minor rewriting
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2976:Inappropriate Title?
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1490:{\displaystyle n}
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3094:Clément Pillias
3084:`De Rham curve´
3034:
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2978:
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2911:Clément Pillias
2891:
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2300:
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2276:
2261:
2256:
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2210:
2205:
2204:
2176:
2175:
2130:
2125:
2124:
2079:
2074:
2073:
2029:
2024:
2009:
2004:
1989:
1984:
1954:
1949:
1925:
1920:
1899:
1894:
1886:
1885:
1861:
1842:
1829:
1815:
1814:
1727:
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1608:Clément Pillias
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107:systems science
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32:on Knowledge's
29:
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11:
5:
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2016:
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2007:
2003:
1996:
1992:
1987:
1983:
1980:
1977:
1974:
1971:
1968:
1961:
1957:
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1948:
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389:
312:
309:
282:
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256:
250:
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230:the discussion
217:
216:
200:
188:
187:
179:
167:
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163:
162:
155:
145:
144:
137:Mid-importance
133:
127:
126:
124:
94:
93:
77:
65:
64:
62:Mid‑importance
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
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2982:De Rham curve
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2934:
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2923:
2922:
2921:
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2131:
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2101:
2098:
2092:
2084:
2080:
2070:
2053:
2044:
2041:
2034:
2030:
2025:
2021:
2014:
2010:
2005:
2001:
1994:
1990:
1985:
1978:
1972:
1959:
1955:
1950:
1946:
1943:
1940:
1937:
1930:
1926:
1921:
1917:
1914:
1911:
1904:
1900:
1895:
1866:
1862:
1858:
1855:
1852:
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1627:
1617:
1613:
1609:
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1601:
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1389:
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1324:
1316:if q≤1/2 and
1298:
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1291:
1282:
1278:
1274:
1269:
1265:
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1238:
1215:
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1096:
1092:
1069:
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1038:
1015:
1011:
1002:
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999:
994:
992:to the plane,
976:
973:
970:
940:
936:
928:
900:
896:
888:
862:
858:
849:
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841:
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2962:67.198.37.16
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2071:
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1628:
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359:
356:
353:
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344:
342:
337:
333:
331:
328:
325:
322:
319:
317:
314:
295:81.99.106.40
284:
260:Low-priority
259:
219:
185:Low‑priority
159:Chaos theory
136:
96:
40:WikiProjects
289:—Preceding
235:Mathematics
226:mathematics
182:Mathematics
30:Start-class
3127:Categories
1003:introduce
379:introduce
2174:(and not
3032:, where
329:given by
303:contribs
291:unsigned
262:on the
139:on the
112:Systems
103:systems
59:Systems
343:Here,
281:Thanks
36:scale.
1725:, so
1629:With
349:first
345:first
334:where
3114:talk
3098:talk
2966:talk
2915:talk
2786:and
2470:and
2123:and
1684:and
1612:talk
1257:by:
367:talk
363:Daqu
299:talk
105:and
254:Low
131:Mid
3129::
3116:)
3100:)
3090:?
3043:∈
3040:ω
3020:ω
3011:ω
3005:−
2993:ω
2968:)
2917:)
2855:−
2406:.
2311:→
2274:→
2069:.
2045:⋯
2042:∘
2022:∘
2002:∘
1947:⋯
1944:∘
1938:∘
1918:∘
1912:∘
1856:⋯
1795:−
1760:∘
1754:∘
1710:−
1614:)
1361:−
1084:,
1030:,
788:−
768:∞
753:∑
620:⋯
617:∘
597:∘
594:⋯
591:∘
571:∘
460:,
406:,
369:)
340:"
336:o
301:•
3112:(
3096:(
3066:)
3063:2
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2996::
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2932:z
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2861:)
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2845:/
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2830:=
2826:)
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2733:/
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2718:=
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2227:)
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2182:R
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2153:T
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2147:)
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2085:0
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2054:1
2051:(
2048:)
2035:3
2031:a
2026:S
2015:2
2011:a
2006:T
1995:1
1991:a
1986:S
1982:(
1979:=
1976:)
1973:1
1970:(
1967:)
1960:n
1956:a
1951:S
1941:R
1931:2
1927:a
1922:S
1915:R
1905:1
1901:a
1896:S
1892:(
1872:]
1867:n
1863:a
1859:,
1853:,
1848:2
1844:a
1840:,
1835:1
1831:a
1827:;
1824:0
1821:[
1801:)
1798:x
1792:2
1789:(
1785:/
1781:1
1778:=
1775:)
1772:x
1769:(
1766:)
1763:R
1757:S
1751:R
1748:(
1745:=
1742:)
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1736:(
1733:T
1713:x
1707:1
1704:=
1701:)
1698:x
1695:(
1692:R
1672:)
1669:1
1666:+
1663:x
1660:(
1656:/
1652:x
1649:=
1646:)
1643:x
1640:(
1637:S
1610:(
1582:n
1578:L
1550:n
1546:2
1541:/
1537:k
1533:p
1510:n
1506:L
1485:n
1474:.
1462:)
1457:0
1453:p
1449:(
1444:1
1440:d
1436:=
1433:)
1428:1
1424:p
1420:(
1415:0
1411:d
1407:=
1402:2
1398:/
1394:1
1390:p
1369:)
1364:1
1358:q
1355:2
1351:p
1347:(
1342:1
1338:d
1334:=
1329:q
1325:p
1304:)
1299:q
1296:2
1292:p
1288:(
1283:0
1279:d
1275:=
1270:q
1266:p
1243:q
1239:p
1216:n
1212:2
1207:/
1203:k
1200:=
1197:q
1186:,
1174:)
1169:0
1165:p
1161:(
1156:1
1152:d
1148:=
1145:)
1140:1
1136:p
1132:(
1127:0
1123:d
1111:,
1097:1
1093:p
1070:0
1066:p
1043:1
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1016:0
1012:d
980:]
977:1
974:,
971:0
968:[
946:)
941:k
937:b
933:(
929:p
906:)
901:k
897:b
893:(
889:x
868:)
863:0
859:p
855:(
850:1
846:d
842:=
839:)
834:1
830:p
826:(
821:0
817:d
791:k
784:2
778:k
774:b
763:1
760:=
757:k
749:=
744:)
739:k
735:b
731:(
727:x
715:,
701:)
696:k
692:b
688:(
684:p
661:)
656:k
652:b
648:(
644:c
610:k
606:b
601:d
584:2
580:b
575:d
564:1
560:b
555:d
551:=
546:)
541:k
537:b
533:(
529:c
517:,
503:k
499:b
487:,
473:1
469:p
446:0
442:p
419:1
415:d
392:0
388:d
365:(
318:"
297:(
266:.
161:.
143:.
109:.
42::
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