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The chemical space from which the periodic
system arose
Wilmer Leal1,2, Eugenio J. Llanos1,2,3,4, Peter F. Stadler1,2,5,6,7,
J¨urgen Jost2,7& Guillermo Restrepo2,5
August 21, 2019
1Bioinformatics Group, Department of Computer Science, Universit¨at Leipzig,
H¨artelstraße 16-18, 04107 Leipzig, Germany
2Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103
Leipzig, Germany
3Fundaci´on Instituto de Inmunolog´ıa de Colombia (FIDIC), Avenida 50 No.
26-20, 111321 Bogota, Colombia
4Corporaci´on SCIO, Calle 57b 50-50 bloque d22 of. 412, 111321 Bogota, Colom-
bia
5Interdisciplinary Center for Bioinformatics, Universit¨at Leipzig, H¨artelstraße
16-18, 04107 Leipzig, Germany
6Institute for Theoretical Chemistry, University of Vienna, W¨ahringerstraße
17, 1090 Vienna, Austria
7The Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501
Aiming at presenting mid 19th century chemical knowledge to chemistry
students, Meyer and Mendeleev embarked on writing textbooks from which
their periodic systems arose [1]. These systems, as stated by Mendeleev [2],
were based on similarity and order relationships among elements, which underlie
the mathematical structure of every possible system [3]. Meyer and Mendeleev
addressed similarity and order using the available chemical compounds by the
1860s, which constitute the chemical space at that time. Here we explore the
size and diversity of this space and its influence upon the periodic system.
Chemical space: provider of order and similarity
In the 1860s chemists ordered elements by their atomic weights. Mendeleev, in
his second 1869 publication about the system, discussed the pros and cons of
selecting other properties as ordering criterion, e.g. electrochemical properties,
relative affinities and valency [4]. Mendeleev settled on atomic weight, mainly
based upon its invariability across all substances containing a particular element.
Compounds were central for atomic weight determinations, which resulted from
finding the smallest common weight of a large set of compounds containing the
reference element. Similarity among elements was addressed through chemi-
cal resemblance, which for Meyer was captured through common valency in
compounds. Mendeleev commented upon physical properties such as optical,
electrical or magnetic ones and vapour densities [4], but discarded them for their
variability over compounds of the same element. He was after a property related
to the essential presence of an element in its compounds, which moved him to
analyse chemical properties such as acidity or alkalinity of oxides. However,
they were ruled out because of the several amphoteric oxides. Mendeleev ended
up taking substance composition as the proxy for similarity, which boils down
to the valency, selected by Meyer. “Thus the fluorine group contains elements
which preferentially combine with a single atom of hydrogen, the oxygen group
with two, the nitrogen group with three, and the carbon group with four atoms
of hydrogen or chloride” [4], Mendeleev claimed. Thus, compounds provided
order and similarity to the system.
The chemical space by 1869
As compounds were central for developing the system, the question that arises
is which compounds and how many were known by the time the system arose,
which prompted us to explore the size of the chemical space and its diversity.
How large was the chemical space?
Compounds can be either extracted from plants, animals and other sources,
or synthesized, or both. They are then reported in the scientific literature,
along with several of their properties. Reaxys c
1is a large database of chemical
information built from the Gmelin and Beilstein Handbooks and the Patent
Chemistry Database, which is a suitable source of information for historical
studies of chemistry [5] as it contains detailed data on reactions and compounds
since 1771, along with information on the associated publications.
Meyer’s and Mendeleev’s first publications on the system date back to 1864
and 1869, respectively. After Mendeleev’s one [2] (March 1st in the Gregorian
calendar, February 17th in the Russian old style calendar) [6], Meyer published a
paper updating his 1864 results [7]. As the spread of the 1869 literature was not
that rapid at those times, and the production of new substances rose exponen-
tially [5], scientists were hardly aware of the most recent results. For example,
Mendeleev wrote in his 1869 extended publication about the system: “So far,
bismuth has not been combined with hydrogen as have the elements similar to
it” [4]. However, BiH3had been synthesized in 1843 [8]. For Mendeleev, ele-
ments similar to Bi were U, Sb, As, P and N (Table 1), which by 1869 had known
hydrides of similar compositions: SbH3[9], AsH3[10], PH3[11] and NH3(UH3
1Copyright c
2019 Elsevier Limited except certain content provided by third parties.
Reaxys is a trademark of Elsevier Limited. Used under license via the Elsevier R&D Collab-
oration Network.
was reported in 1949 [12]). Moreover, Mendeleev acknowledged little knowledge
about In [4], but there were already reports of more than 20 of its compounds.
The question that arises is how knowledge, or ignorance, of the chemical
space may have affected the periodic system. We addressed the question by
retrieving information from Reaxys from 1771 up to 1868 (included), which
amounts to 26,502 single-step reactions and 11,484 substances, mainly reported
in Gmelins Handbuch der anorganischen Chemie and gathered from leading
19th century journals, such as Justus Liebigs Annalen der Chemie,Journal f¨ur
praktische Chemie,Annalen der Physik,Annales de Chimie,Comptes Rendus
Hebdomadaires des Seances de l’Academie des Sciences,Jahresbericht ¨uber die
Fortschritte der Chemie und Verwandter Theile Anderer Wissenschaften and
Journal of the Chemical Society among others. These compounds span 60 ele-
ments: H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl, K, Ca, Ti, V, Cr, Mn,
Fe, Co, Ni, Cu, Zn, As, Se, Br, Rb, Sr, Zr, Nb, Mo, Ru, Rh, Pd, Ag, Cd, In, Sn,
Sb, Te, I, Cs, Ba, La, Ce, Ta, W, Os, Ir, Pt, Au, Hg, Tl, Pb, Bi, Th, U. Meyer’s
1870 paper does not include H, La, Ce, Th and U [7], whereas Mendeleev’s first
1869 publication included them plus Er, Yt and Di [2]. Er and Yt, along with
In, were elements whose identity was questioned by Mendeleev and expressed as
?Er, ?Yt and ?In in his table. Yt was the symbol used until 1920 for Y [13] and
the first Y (or Yt) reaction is from 1872. Thus, neither Meyer nor Mendeleev
had clear information about the element. Er was also problematic. By 1868 it
was ignored that Er was actually a mixture of an element later (1878) coined
Er and Yb, which were separated one year later into the current Er, Ho and
Tm; and Sc and a Yb, respectively [14]. In 1907 the 1879 Yb was found to be a
mixture of the current Lu and Yb [14]. Moreover, there was confusion between
erbium and terbium on the basis of their spectra [14] and it is now known that
Di, reported by Mendeleev as a chemical element, is a mixture of Pr and Nd.
Therefore, we excluded Er, Yt and Di from our analysis and all the remaining
study is based on our findings for the aforementioned 60 elements.
To know the extent of the chemical space populated by the different ele-
ments, we counted the number of compounds containing each element (Figure
1a; Supplementary, Table 1), where the enormous amount of O, H, C and N
compounds is evident. Were O compounds absent, only 26% of the space would
remain. If H substances were also removed, the available space would be 11.4%.
By ignoring compounds containing O, H or C, the spanned space would be
only 6% of that of 1869, which drops to 5% if N substances were disregarded.
The importance of these organogenic elements contrasts with the percentage of
space spanned if other quartets of elements were deleted. For example, remov-
ing halogen compounds would have much less dramatic effects, as 64% of the
space would remain. Disregarding elements with few compounds such as Ta,
Ce, Th and La would allow access to 96% of the space.
Thus, the chemical space was rather unevenly explored by 1869, oxygen and
hydrogen being the elements with most compounds. Chemists were aware of
this, as Mendeleev wrote: “The most widely distributed substances in nature
possess the smallest atomic weights” [4] and “The higher atomic weights belong
to elements that are rarely encountered in nature, which do not form large
deposits, and which have, as a consequence, been little studied” [4].
As elements populate the space through combinations with others, we won-
dered how distributed the combinations of elements were.
Diversity of the chemical space: combinations of elements
By a combination we mean the elements present in a compound and arranged
in lexicographic order, e.g. HOS for H2SO4[5]. We found 3,022 combinations
and analysed the distribution of their sizes (number of elements). Was the
space populated with compounds of several elements? Or with substances of
just few elements? For 60 elements, there are 1.15 ×1018 possible combinations
(Methods). We found that chemists reported compounds of few elements: from
two up to eight. Moreover, about three quarters of the combinations were of
three up to five elements (Figure 1b). The differences between experimental
and theoretical number of combinations across the different sizes (Figure 1b),
show that there was still plenty of opportunity to populate the space with new
compounds within the observed range.
To assess whether the space was also concentrated on some few combinations,
we analysed the percentage of chemical space spanned by combinations weighted
by their compounds (Figure 1c). The left hand side of the plot shows that
combinations associated to one compound only, account for about 13% of the
space. The right hand side shows the most frequent combinations: CHO and
CHNO, which account for 8% and 7.5% of the space, respectively. We found
that 37% of the combinations (1,120) account for 80% of the compounds and
only 172 combinations spanned about half of the space. Strikingly, only 6% of
the combinations gathered 50% of the reported substances. The top-20 of these
combinations (Figure 1c) account for 30% of the space and their compositions
show that the most populated combinations are based on organogenic elements
including C, H, O, N, Cl, S, Br; which are among the elements with most
compounds (Figure 1a). Notoriously, top-20 combinations are C and H based.
The minor metallic span of the space is evident through the most populated
metal combination: CHAgNO, which only gathers 0.8% of the space (Figure
1c). This is consistent with recent results on composition and growth of the
chemical space [5].
The number of combinations per element follows a similar trend to that of
compounds per element shown in Figure 1a (Supplementary, Figure 1; Spearman
correlation of 0.99). The question arising is on the ratio between the number
of compounds and of combinations for each element. We calculated it (Figure
1a, inset) and found that organogenic elements have much more compounds
than combinations, when compared with other elements. C has six times more
compounds than combinations and H, O and N, 4.6, 3.8 and 3.9, respectively.
For La and Ta, each one of their combinations correspond to a single compound.
We wondered whether compounds of an element are concentrated in few
of its combinations or spread over them. Figure 1d shows that organogenic
elements concentrate most of their compounds in very few combinations: H,
C, N and O gather three quarters of their compounds in about 1% of their
combinations. This contrasts with the spread of other elements such as V and
Au, which concentrate three quarters of their compounds in an ample range of
combinations; 60% of V combinations and 67% of Au ones.
To explore the role of C combinations and to contrast them with some non-
C combinations, we analysed how often some molecular fragments appear over
time in known substances. The fragments studied were sulphate and nitrate
anions as typical inorganic non-C ensembles. Carbonate anion was studied
as an example of an inorganic C-fragment. Monosubstituted benzenes; pri-
mary amines and carboxylic acids were the organic C-ensembles studied. The
organometallic junction C-M, M being any of the following elements: Zn, Sb,
As, Hg, Tl, Bi, Pb, Rh, Co, Pt, Li, Be, Al, Fe, Si, Ge, was considered (Meth-
ods). Figures 1e-f show that before 1830 the number of compounds containing
each one of the fragments was rather low, not surpassing 29 compounds per year.
However, after 1830 organic fragments surged, while inorganic and organometal-
lic fragments grew but not as dramatically as their organic counterparts. Al-
though we showed elsewhere [5] that synthetic compounds steadily became the
way to reach new substances well before W¨ohler’s 1828 synthesis of urea, the
blossoming of organic compounds observed after 1830 and therefore the bias of
the chemical space towards C-combinations may be caused by W¨ohler’s invig-
orating influence upon synthesis [15]. Before, C-combinations resulted from a
balance of organic, organometallic and inorganic compounds. Therefore, Meyer
and Mendeleev had at their disposal a chemical space with inorganic tradition
and with about 40 years of very rapidly growing organic colonies.
Similarities arising from the chemical space
Granted the ordering of elements by atomic weight, we only need to determine
the similarities among chemical elements arising from the available chemical
space by 1869 to obtain the periodic system at that time [3]. We addressed
similarity using Meyer’s and Mendeleev’s approach based on compounds. Al-
though we keep the spirit of our previous approaches [16], here we consider key
features of chemical similarity that have been disregarded before. As Mendeleev
stated: “The elements, which are most chemically analogous, are characterised
by the fact of their giving compounds of similar form RXn” [17], an example of
such common formula for similar elements is R2O, where R is an alkali metal.
This similarity statement can be interpreted as the degree of replaceability of
element xin the formulae of compounds of yand the converse, whenever xand
yare similar. From compounds of the chemical space, we extracted their differ-
ent formulae. To quantify the similarity of element xregarding element y, we
took each formula containing xand replaced xby the symbol A. The resulting
formula was arranged by lexicographic order. Likewise, ywas replaced by Ain
its formulae, which were lexicographically ordered (Figure 2). The similarity of
xregarding ywas quantified as the fraction of arranged formulae that xshares
with y. This is an asymmetric relation, as shown in Figure 2, where Be is more
similar to Mg (s(Be →Mg)) than the converse (s(Mg→Be)). This occurs as
four of the six Be arranged formulae are shared with Mg, whereas four of the
seven Mg formulae are shared with Be. In fact, by considering the actual space
by 1869, the asymmetry strengthed as s(Be →Mg)=24/49=0.49, while s(Mg →
Be)=24/296=0.081. Thus, in half of Be formulae, the element can be replaced
by Mg and the resulting formulae are part of the space; whereas in less than
10% of Mg formulae, Be can replace Mg.
Figure 3 shows the most similar element for each element according to the
1869 chemical space. There are eight components of similar elements, labelled
according to the oxidation state of their elements. The type of compounds
making elements of each component similar is provided. The yellow component
contains alkali metals and Ag, Tl and H (oxidation state I), which are similar
because of their arsenates, sulphates, nitrates, carbonates, chlorides and iodides.
Alkaline-earth metals group together with other elements having oxidation state
II (blue component), except Si and Zr. Disregarding Si and Zr, with oxidation
state IV, elements of this component form fluorides, chlorides and sulphides.
Elements with oxidation state III forming chlorides gather in the red component,
which contains pnictogens, except N. Halogens (oxidation state -I) show up
together with N and V in the pink component, which form oxygenated carbon
compounds: CH2(OH)V, CH3NCO and acetyl halides. The white component
groups chalcogens and C (oxidation state -II) that are similar because of their
ammonia derivatives: CH3NH2, NH3·H2O, NH4·HS, NH4·HTe and NH4·HSe.
Transition metals are divided into three components, the green one gathering
ferrous metals, Al, In and some other elements with oxidation state III sharing
oxides. Elements of the orange component have oxidation states IV, while those
of the purple component a mixture of oxidation states, represented in several
types of common compounds.
Note that even if elements of a component are connected by sequences of
similarities, it does not follow that two non-adjacent elements of a component
hold a high degree of similarity. This is a consequence of the lack of transitivity
of similarity relations [18]. For example, in the blue component Zr is similar
to Si, as they form compounds where their oxidation state is IV, e.g. XO2,
XCl4. Likewise, Si and Ti are similar, as well as Ti and Sn. However, Ti and
Sn are also similar because of their compounds with oxidation state II. If we
keep following the similarity arrows, the presence of oxidation state II becomes
stronger, while that of IV winds down.
Of the similarity relations (Figure 3), those of organogenic elements are the
weakest ones, as these elements only share less than 3% of their large sets of
arranged formulae with their most similar elements. The strongest similarities
occur for Rb, Cs, La, Se, Br, Li, Be, Pd and Tl, which share more than 50%
of their arranged formulae with their most similar elements. In fact, for only
less than half of the formulae of 85% of the elements, they can be replaced by
their most similar element. Hence, the chemical space by 1869 made most of the
elements to be dissimilar from each other. In such a space of tiny similarities,
chemists, nevertheless, were able to recognise several of them. Was it the result
of chemical genius? Or a consequence of the organogenic bias of the space?
To assess whether deep knowledge of the space was needed to gauge simi-
larities among elements, or, on the contrary, whether partial knowledge sufficed
to detect similarities, we took random samples of the formulae in the space of
different sizes and analysed how often the similarities given by the whole for-
mulae in the space (Figure 3) were present in the samples (Methods). After
analysing the 9,752 formulae spanning the space, we found that there are sta-
ble similarities, still evident in tiny portions of the space, e.g. Br→Cl, Na→K,
O→S, Cl→Br, S→O, K→Na, Se→S, As→P (shown at the top of Figure 4).
For example, Br→Cl is a similarity often observed with even 10% of the space
(about 970 formulae). In contrast, similarities that are only evident by consid-
ering large portions of the formulae are Zn→Mg, Cs→Na, Ca→Ba and those
at the bottom of Figure 4. For instance, Ru→Mo only appears in more than
half of the samples if the formulae analysed account for more than 95% of the
known ones.
Figure 4 shows that most of the stable similarities occur for main group
elements, while transition metal ones require deep knowledge of the space. This
is a consequence of the small number of formulae associated to transition met-
als, formulae that are easily discarded by random sampling (Supplementary,
Table 3). Moreover, by taking less than half of the formulae (4,876), about the
same proportion of similarities is obtained; mainly among main group elements,
having by far more formulae of few combinations than transition metals.
Meyer’s and Mendeleev’s similarities for chemical elements are shown in
Table 1 [19, 2, 4, 7]. Mendeleev explained that in his 1869 table “in certain
parts of the system the similarity between members of the horizontal rows will
have to be considered, but in other parts, the similarity between members of the
vertical columns” [4]. By contrasting Table 1 with Figures 3 and 4, we found
that Meyer’s and Mendeleev’s similarities match to a large extent with the
similarities arising from the chemical space, especially for main group elements.
Their mismatches occur for transition metals, where deep knowledge of the space
was necessary to detect the subtle differences among these elements.
To conclude, our results show the enormous bias of the chemical space by
1869 towards organic compounds, which populated the exponentially growing
space in the last 40 years and tilted the scale of compounds in their favour. Be-
fore evident relevance of organic compounds, the space had a more homogeneous
mixture of inorganic and organic compounds [5].
The size and diversity of the space shows that with knowledge of organogenic
elements, chemists would have gauged a representative part of the space. In fact,
random knowledge of about half of the space would have led to about half of
the similarities resulting from the space, mainly for main group elements. This
justifies the almost invariable resemblances among chemical elements reported
by Meyer and Mendeleev and others [20] and the difficulty in finding stable
similarities for transition metals [16, 21, 22], characterised by few compounds,
therefore requiring a more complete knowledge of the space.
Meyer’s and Mendeleev’s idea of a book concentrating the most salient fea-
tures of mid 19th century chemical knowledge, from which the periodic system
arose, could have concentrated their content on oxoacid salts, halides, sulphides,
simple carbon oxygenated compounds, ammonia derivatives and oxides, which
Table 1: Similarities reported by Meyer and Mendeleev in their 1869-1870 pub-
lications. Bold face sets indicate similarities found using the chemical space by
1869.
Meyer Mendeleev
B, Al, In, Tl Pt, Ir, Os Si, Ti, Zr, Sn
C, Si, Sn, Pb Cu, Ag Mg, Zn, Cd
Ti, Zr K, Rb, Cs B, Al
N, P, As, Sb, Bi Te, Se, S Ca, Sr, Ba
V, Nb, Ta Pb, Ba, Sr, Ca Li, Na, K
O, S, Se, Te Tl and alkali metals Bi, Sb, As, P, N, U
Cr, Mo, W Cl, Br, I Pd, Rh, Ru
F, Cl, Br, I O, S, Se, Te N, P, As, Sb
Mn, Fe, Co, Ni Ag, Pb, Hg P, As, Sb
Ru, Rh, Pd V, Mo, W Ta, Sn, Ti
Os, Ir, Pt V, Nb, Sb Fe, Ce, Pd, Pt
Li, Na, K, Rb, Cs C, B, Si, Al Ba, Pb, Tl
Cu, Ag, Au V, Cr, Nb, Mo, Ta, W U, B, Al
Be, Mg, Ca, Sr, Ba H*Ni, Co
Zn, Cd, Hg
∗H is only slightly similar to K, only about 2% of its arranged formulae are
shared with K.
are the carriers of most of the similarities among chemical elements. The chal-
lenge is writing the book of chemistry for the contemporary available space and
to explore the influence of the current space upon the periodic system. Is it still
there?
Acknowledgements
W.L. acknowledges support from the German Academic Exchange Service (DAAD):
Forschungsstipendien-Promotionen in Deutschland, 2017/2018 (Bewerbung 57299294).
Contributions of authors
G.R. conceived the idea; W.L. and G.R. designed the research; W.L. dumped
and analysed data; W.L., E.J.L., P.F.S. and G.R. devised similarity measure;
W.L. and E.J.L. computed and analysed similarities; W.L., E.J.L., P.F.S., J.J.
and G.R. discussed the results; G.R. wrote the original draft; W.L., E.J.L.,
P.F.S., J.J. and G.R. reviewed and edited the original draft.
Supplementary information
•Figure 1: Number of combinations per element
•Figure 2: Structures of the fragments studied
Methods
Number of combinations.
It was calculated as P60
i=2 60
i. This is a rough upper bound disregarding va-
lency and compound stability.
Molecular fragments.
Search of these fragments was performed by exploring the connection tables
of the compounds. A connection table is a “listing of atoms and bonds, and
other data, in tabular form” [23]. For the sake of clarity, C-Ameans that at
least a bond between C and Ais reported on the table. It does not necessarily
mean that C and Aare bonded by a single covalent bond. The structures of the
analysed molecular fragments are shown in Figure 2 of Supplementary Material.
Sampling the space.
We randomly took s% of the space and determined the most similar element of
each element. This experiment was carried out 100 times. For each similarity
A→Bresulting for the whole space (Figure 3), we counted its frequency
of appearance in the 100 experiments of size s%. The fractions of the space
analysed ranged from 95%, 90%, 85%,. . ., 5%. The higher the frequency of
appearance of the similarities shown in Figure 3 in the 100 experiments of each
size s, the more stable the similarity regarding size sis. Moreover, the higher
the frequency for different values of s, the more size independent the similarity
is.
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W.L. acknowledges support from the German Academic Exchange Service (DAAD):
Forschungsstipendien-Promotionen in Deutschland, 2017/2018 (Bewerbung 57299294).
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8
10
12
14
16
18
20
a)
H
C
O
N
S
Atomic number
Percentage
1 10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
H
C
N
O
Atomic number
Ratio
—
2-ary
3-ary
4-ary
5-ary
6-ary
7-ary
8-ary
100
101
102
103
104
105
106
107
108
109
1010
b)
n-ary combinations
Number of combinations
Experimental Theoretical
100101102103
100
101
c) CHNO
CHN
CHO
CHOS
CHClNO
CHClO
CHNOS
CHCl
CH
CHBr
CHBrNO
CHCuO
CHNaO
CHNS
CHNOPb
CHClN
CHIN
CHKO
CHS
CHAgNO
Number of compounds
Percentage
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 6972 77 81 85 89 93
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d)
Atomic Number
Percentage of combinations
1800 1810 1820 1830 1840 1850 1860 1870
0
10
20
30
40
50
60
70
80
e)
year
Number of compounds
CO2−
3
SO2−
4
NO−
3
C-M
1800 1810 1820 1830 1840 1850 1860 1870
0
50
100
150
200
250
300
350
400
450
f)
year
Number of compounds
NH2-R
R-COOH
R-C6H5
Figure 1: Size and diversity of the chemical space: a) Chemical elements known
by 1869 and the percentages of compounds in which they take part. These per-
centages are non-additive as a single compound adds to each one of its elements,
e.g. H2O contributes to both H and O counting. Inset) Ratio between the num-
ber of compounds and of combinations for each element. b) Size distribution
of combinations. Number of n-ary reported combinations of elements (black)
and theoretical bounds (gray) (Methods). c) A pair (x, y) indicates that y=cx
percentage of the chemical space is spanned by ccombinations accounting for x
compounds. d) Box plots of the distribution of compounds per combination for
each element, with whiskers indicating combinations with least and most popu-
lated compounds; red line shows the median of each distribution. e) Temporal
distribution of several typical inorganic, organometallic and f) organic molecu-
lar fragments. C-M stands for the bond C-metal, with M={Zn, Sb, As, Hg, Tl,
Bi, Pb, Rh, Co, Pt, Li, Be, Al, Fe, Si, Ge}.
Chemical space
BeBr2MgBr2
Be3(AsS3)2BeCO3MgCO3
Be2ZrO4Be3As2Mg3As2
BeSO4MgSO4
MgHPO4
CH3MgI
Mg2SiO6
Arranged formulae
A3As2S6
A2O4Zr
FBe FMg
s(Be →Mg)=|F
Be
∩F
Mg
|
|F
Be
|=4
6
s(Mg→Be )=|FBe∩FMg|
|FMg|=4
7
ABr2
ACO3
A3As2
AO4S
AHO4P
ACH3I
A2O6Si
Figure 2: Similarity among chemical elements. A toy-chemical space of 13
formulae. Each formula provides an arranged formula for an element in the
given formula. For example, A2O4Zr is the arranged formula of Be from formula
Be2ZrO4. Arranged formulae of element xare gathered in Fx. Similarity of
element xregarding element yis given by s(x→y).
Ag
Al
As
Au
B
Ba
Be
Bi Br
C
Ca Cd
Ce
Cl
Co
Cr
Cs Cu
F
Fe
H
Hg
I
In
Ir
K
La
Li
Mg
Mn
Mo
N
Na Nb
Ni
O
Os
P
Pb
Pd
Pt
Rb
Rh
Ru
S
Sb
Se
Si
Sn
Sr
Ta
Te
Th
Ti
Tl
U
V
W
Zn
Zr
Figure 3: Most similar elements by 1869. Arrows A→Bindicate that Ais
most similar to B. Node size is proportional to the number of formulae in which
the element represented by the node is involved. Arrow size is proportional to
the similarity s(A→B). Shared formulae for elements belonging in a compo-
nent (connected set of elements) are shown. Formulae of the blue component
disregard Zr and Si. Likewise, disulfide of elements in the orange component
does not apply to Th and Ce.
Figure 4: Stability of similarities regarding chemical space size. Most similar
element for each element obtained by considering the whole space and their fre-
quency of appearance (colour scale) in random samples of the space of different
sizes (ranging from 5% to 95% of the space).
