<p>(<b>A</b>) Results of maximum parsimony (MP hereinafter) analyses of the conventional plastid genomic DNA matrix of the bamboos (Arundinarieae, Poaceae, flowering plants) from [<a href="#B28-dna-05-00010" class="html-bibr">28</a>]. Final trees were rooted relative to <span class="html-italic">Dendrocalamus latiflorus</span> Munro [<a href="#B28-dna-05-00010" class="html-bibr">28</a>]. The cladogram represents the median consensus tree based on Robinson–Foulds (RF) distance (with the best score found = 8837) of 184 shortest output trees of length = 5019 (CI = 0.89, RI = 0.91). The number of taxa = 157. All constant characters from the original alignment are excluded from the analysis. The number of variable characters = 4304, number of parsimony-informative characters = 2003. * nodes received MP Jackknife (JK) support > 50% after 20,000 fast JK replicates; ! nodes recovered MP Bootstrap support in the analysis from [<a href="#B28-dna-05-00010" class="html-bibr">28</a>] (200 full heuristic replicates). (<b>B</b>) Results of MP of the binary representation of the conventional DNA matrix from A., re-coded following the proposed <span class="html-italic">1001</span> Method 1. Initial binary data were polarized before analysis relative to <span class="html-italic">D. latiflorus</span>, assumed as an outgroup [<a href="#B28-dna-05-00010" class="html-bibr">28</a>]. The cladogram represents the majority-rule consensus of 191 shortest output trees of length = 10,014 (CI = 0.88, RI = 0.89). The number of taxa = 157. The number of binary characters = 8783, number of parsimony-informative characters = 4088. * nodes received MP Jackknife (JK) support > 50% after 20,000 fast JK replicates. (<b>C</b>) Results of MP analyses of the binary representation of the conventional DNA matrix from A., re-coded following the proposed <span class="html-italic">1001</span> Method 2. Data polarized before analysis relative to <span class="html-italic">D. latiflorus</span>, assumed as an out-group based on the previous results of [<a href="#B28-dna-05-00010" class="html-bibr">28</a>]. The cladogram represents the majority-rule consensus of 139 shortest output trees of length = 4993 (CI = 0.89, RI = 0.91). The number of taxa = 157. The number of binary characters = 4993, number of parsimony-informative characters = 2027. * nodes received MP Jackknife (JK) support > 50% after 20,000 fast JK replicates. All MP analyses were conducted using program PAUPrat [<a href="#B29-dna-05-00010" class="html-bibr">29</a>,<a href="#B30-dna-05-00010" class="html-bibr">30</a>,<a href="#B31-dna-05-00010" class="html-bibr">31</a>] as implemented in CIPRES [<a href="#B32-dna-05-00010" class="html-bibr">32</a>] following 200 ratchet replicates with no more than 10 trees of length greater than or equal to 1 saved in each replicate, and the TBR branch swapping/MulTrees option in effect; -pct = 20%, all characters weighted uniformly, and gaps were treated as ‘‘missing”. MP jackknifing [<a href="#B33-dna-05-00010" class="html-bibr">33</a>] was conducted using PAUP* version 4.a168 [<a href="#B31-dna-05-00010" class="html-bibr">31</a>] (PAUP* hereinafter) as implemented in CIPRES [<a href="#B32-dna-05-00010" class="html-bibr">32</a>]. Robinson–Foulds consensus [<a href="#B14-dna-05-00010" class="html-bibr">14</a>,<a href="#B34-dna-05-00010" class="html-bibr">34</a>] calculated using RFS version 2.0 [<a href="#B34-dna-05-00010" class="html-bibr">34</a>]. Majority-rule consensus calculated in PAUP* [<a href="#B31-dna-05-00010" class="html-bibr">31</a>]. Branches with a minimum length of zero collapsed. All gaps and ambiguities of the conventional DNA matrix (<b>A</b>) were recoded as missing data (“?”) before binary permutations. Roman numerals correspond to the “major lineages” of Arundinarieae [<a href="#B28-dna-05-00010" class="html-bibr">28</a>].</p> Full article ">Figure 2
<p>The results of two three-taxon statement analyses (3TA hereinafter) of Clades 1 and 2 (<a href="#dna-05-00010-f001" class="html-fig">Figure 1</a>). The DNA alignments have been polarized following <span class="html-italic">1001</span> Method 2 and subsequently established as binary three-taxon matrices using TAXODIUM version 1.2 [<a href="#B18-dna-05-00010" class="html-bibr">18</a>] (TAXODIUM hereinafter). Following the results of the previous analyses (<a href="#dna-05-00010-f001" class="html-fig">Figure 1</a>), <span class="html-italic">Indocalamus wilsonii</span> (Rendle) C.S.Chao and C.D.Chu (Clade 1) and <span class="html-italic">Bergbambos tessellata</span> (Nees) Stapleton (Clade 2) were assumed to be outgroup taxa before Method 2 was applied to the DNA characters. (<b>A</b>) The results of the first 3TA (Clade 1). Majority-rule consensus of 193 shortest output trees of length = 527,046 (CI = 0.92, RI = 0.91). The number of taxa in the 487168 character–3TA matrix is 72. All 487,168 3TSs are parsimony-informative and weighted uniformly. (<b>B</b>) The results of the second 3TA (Clade 2). Majority-rule consensus of 201 shortest output trees of length = 187,857 (CI = 0.86, RI = 0.83). The number of taxa in the 161,027 character–3TA matrix is 80. All 1,610,278 3TSs are parsimony-informative and weighted uniformly. For the meaning of Roman numerals and the details of the MP analyses, see the legend of <a href="#dna-05-00010-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 3
<p>(<b>A</b>) The simplified phylogeny of flowering plants and outgroups resulted from the MP analysis of the 38,553 bp cpDNA alignment from [<a href="#B35-dna-05-00010" class="html-bibr">35</a>]. The general strategy of the analysis is described in [<a href="#B18-dna-05-00010" class="html-bibr">18</a>]. The heuristic search for the most parsimonious tree was performed with the implied weights [<a href="#B36-dna-05-00010" class="html-bibr">36</a>] included in the search procedure, and the value of the <span class="html-italic">k</span>-function was assigned as three. The phylogeny is established as a single phylogram. Goloboff fit = −10,023.39940, with the actual length of the tree equal to 48186, CI = 0.55, RI = 0.61. The number of informative characters is equal to 13,328. (<b>B</b>) The most parsimonious hierarchy of patterns was obtained from the MP analysis of the same strategy as in (<b>A</b>). The latter was based on the polarized binary matrix recoded from the conventional cpDNA alignment (<b>A</b>) following <span class="html-italic">1001</span> Method 1, with <span class="html-italic">Cryptomeria</span> (Cupressaceae Bartlett, gymnosperms) assumed as the best outgroup. The hierarchy of patterns is established as a single cladogram. Goloboff fit = −24165.80162 with the actual length of the tree equal to 102,724, CI = 0.49, RI = 0.60. The number of informative characters equals 32,141. (<b>C</b>). The most parsimonious hierarchy of patterns resulted from the MP analysis, which followed the same strategy as in A (see above) but without implied weights [<a href="#B36-dna-05-00010" class="html-bibr">36</a>] included in the search procedure. The analysis was based on the polarized binary matrix recoded from the conventional cpDNA alignment (<b>A</b>) following <span class="html-italic">1001</span> Method 2, assuming <span class="html-italic">Cryptomeria</span> as the best outgroup. The hierarchy of patterns is established as a single cladogram of the length 48,552, CI = 0.56, RI = 0.62. The number of informative characters equals 15,653. (<b>D</b>) The single most parsimonious hierarchy of patterns resulted from the MP analysis, which followed the same strategy as in (<b>A</b>) (see above) but without implied weights [<a href="#B36-dna-05-00010" class="html-bibr">36</a>] included in the search procedure. The analysis was based on the three-taxon statement matrix with 1,652,888 fractionally weighted [<a href="#B4-dna-05-00010" class="html-bibr">4</a>,<a href="#B12-dna-05-00010" class="html-bibr">12</a>,<a href="#B14-dna-05-00010" class="html-bibr">14</a>,<a href="#B18-dna-05-00010" class="html-bibr">18</a>] three-taxon statements calculated by TAXODIUM [<a href="#B18-dna-05-00010" class="html-bibr">18</a>]. This matrix is derived from the polarized binary representation (<span class="html-italic">1001</span> Method 2) of the 28,196 bp largest clique, estimated by PHYLIP version 3.695 [<a href="#B19-dna-05-00010" class="html-bibr">19</a>] based on a 38,553 bp cpDNA alignment (<b>A</b>). <span class="html-italic">Cryptomeria</span> is assumed to be the best outgroup. The hierarchy of patterns is established as a cladogram of the length of 230,181.7318, CI = 0.99, RI = 0.99. The number of informative characters (three-taxon statements) equals 1 652 888.</p> Full article ">Figure 4
<p>(<b>A</b>) An unrooted simplified molecular phylogeny of <span class="html-italic">Ceratophyllum</span> (Ceratophyllaceae A. Gray, flowering plants) [<a href="#B37-dna-05-00010" class="html-bibr">37</a>], showing the ambiguous placement of <span class="html-italic">C. echinatum</span> [<a href="#B37-dna-05-00010" class="html-bibr">37</a>]. (<b>B</b>) A summary of the cladistic analyses [<a href="#B38-dna-05-00010" class="html-bibr">38</a>], demonstrating that <span class="html-italic">C. echinatum</span> is a sister group to the narrowly defined genus <span class="html-italic">Ceratophyllum</span>. All analyses (<b>B</b>) were based on the binary ’presence–absence’ representation of the molecular data from [<a href="#B37-dna-05-00010" class="html-bibr">37</a>], adding an artificial all-zero outgroup. As a result of the cladistic analyses of the binary recoded DNA sequence data [<a href="#B37-dna-05-00010" class="html-bibr">37</a>,<a href="#B38-dna-05-00010" class="html-bibr">38</a>], <span class="html-italic">C. echinatum</span> was defined as a sister group of the narrowly circumscribed genus <span class="html-italic">Ceratophyllum</span> [<a href="#B38-dna-05-00010" class="html-bibr">38</a>] and transferred to the newly established genus <span class="html-italic">Fassettia</span> based on the obtained phylogenetic placement [<a href="#B38-dna-05-00010" class="html-bibr">38</a>]. See [<a href="#B38-dna-05-00010" class="html-bibr">38</a>] for details of the cladistic analyses and taxonomic treatment. Clade “<span class="html-italic">Ceratophyllum</span>” is marked with an asterisk (*). This figure also shows the ‘presence–absence’ binary coding (<b>B</b>) of the DNA sequence data (<b>A</b>), as implemented in <span class="html-italic">1001</span>.</p> Full article ">Figure 5
<p>Leibniz’s original four-digit binary representation of Arabic numbers one, two, four, and eight (indicated by exclamation marks, added by us). In the third column of this table, Leibniz himself linked this representation with the combination of solid and dotted lines, each corresponding to one of the four <span class="html-italic">T’ai Hsüan Ching</span> tetragrams (indicated by exclamation marks, added by us), namely the tetragrams <span class="html-italic">Penetration</span>, <span class="html-italic">Legion</span>, <span class="html-italic">Fullness</span>, and <span class="html-italic">Law</span> (<span class="html-italic">Model</span>) [<a href="#B73-dna-05-00010" class="html-bibr">73</a>]. Reproduced from Leibniz’s manuscript <span class="html-italic">De Dyadics</span>, as interpreted and translated by Yakovlev [<a href="#B72-dna-05-00010" class="html-bibr">72</a>], see pp. 195, 201, and 202.</p> Full article ">