Linking narratives of the periodic table to Enlightenment pragmatism and Taoist mysticism

Linking narratives of the periodic table to Enlightenment pragmatism and Taoist mysticism

Introduction

The format of the periodic table at any moment in time reflects the understandings and preferences of the chemists of the day. To that end, it has been dynamic since its inception. There continues to be debate about the appropriate position of a few elements, and whether that position should be driven by the similarity (or difference) of an element’s chemical behaviour to (or from) other elements,1 or whether it should be driven by physical attributes (e.g. the size of atoms and ions)2 or by quantum-mechanical considerations.

In pictorial representations of the table, there is an expectation that the sequence of elements must be in order of increasing atomic number, but beyond that the scope is potentially large. Scerri3 draws attention to three particularly visually striking examples: Benfey’s diagram which has “bulges for the transition metals that point outwards from the main circular system”,3 a three-dimensional table that “provides continuous or pseudo continuous representation”,4 and “a periodic galaxy [that] emphasises the continuity of the elements unlike the majority of periodic tables that imply breaks in continuity between successive periods”.5

These representations rightly highlight the location of elements relative to each other, but, of course, each chemical element and its compounds also has its own history of discovery, its own properties, uses, and its own economic, medical and social values. These ‘facts’ have been compiled in various ways in numerous descriptive books about the periodic table; this paper uses the 2018 version of Albert Stwertka’s A Guide to the Elements as its prime example, and refers to the book’s commentaries on individual elements as ‘vignettes’. Faced with the large amount of information about any given element, any compiler of such vignettes is required to choose between the information to include or omit, in order to ensure the “making sense of what otherwise might be a chaotic jumble of facts about the elements and their many molecular combinations”.6

However rational compilers assert their choice to be, these are choices informed not only by their understanding of the periodic table per se, but also by their own experiences, as well as their prejudices and preconceptions. There may even be constraints imposed by the publisher of the vignettes in order to ensure the envisaged sales of the book are not compromised.  This may mean that – despite skepticism7– the apparently rational scientific world of the periodic table could become more characteristic of the ‘natural world’: “… a world of “infinite varieties and complexities, a multi-dimensional world, which contains no straight lines or completely regular shapes, where things do not happen in sequence, but all together.”8

Pragmatism versus mysticism

In addition to the purely factual aspects of the chemical elements, stories can be woven around an element focusing on the scientists involved in their initial discovery of the elements or in the elements’ subsequent use, for which this analysis uses the autobiographically-based stories related to Primo Levi’s experiences as a practising chemist,9 and two versions of Sam Kean’s The Disappearing Spoon…: the standard one10 and that modified for younger readers.11 In all three books, the stories go beyond mere observations, and involve – to a greater or lesser extent – the participants in the discovery of the elements and those involved in the development of technologies that use them. This merging of the observer and observed, i.e., the merging of subject and object, could be considered to give the narrative something of a mystical character, as suggested below:

“The idea of ‘participation instead of observer’ … is an idea which is well known to any student of mysticism. Mystical knowledge can never be obtained just by observation, but only by full participation with one’s whole being. The notion of the participator is thus critical to the Eastern world view, and the Eastern mystics have pushed this notion to the extreme, to a point were observer and observed – subject and object – are not only inseparable, but also become indistinguishable.”12

An elaboration of this notion from a Taoist perspective is:

“The Taoist relaxes the body, calms the mind, loosens the grip of categories made habitual by naming, frees the current of thought for more fluid differentiations and assimilations, and instead of pondering choices lets the problems solve themselves as inclination spontaneously finds its own direction.”13

A story developed in this way might differ considerably from a fact-based narrative, because: “while the Taoists were extremely interested in Nature, they tended to distrust reason and logic so that the writings of the Tao tended to remain somewhat inscrutable…”14

The notion that “the mystic ‘constructs’ [italics added] the framework of mystical encounter, or experiences ‘intended’ at the outset of the mystical path”15 stands in contrast to Western science’s “Enlightenment structures [which] require us to create a dispassionate distance and separation from what we perceive”.16 In a somewhat trivial illustration of this requirement of distancing, only recently – and still only in comparatively few scientific journals – do authors refer to themselves in the first person. In a research context, in the late 20th century, there was a suggestion that children’s learning was not achieved by filling vacant minds, but that learners ‘constructed’ their knowledge as a combination of information that they already had, augmented by what they were formally taught,17 from which it could be inferred that no knowledge is impartial, but that it is all constructed. When applied to science, this idea was vehemently opposed by some science educators,18 but the prospect of a link between what came to be referred to as ‘constructivism’ and Eastern mysticism has been raised again more recently by,19 supported, albeit with some criticism,20 echoing an earlier perspective of what might happen at the limit of rational and intuitive knowledge:

“Our faltering mind must then seek repose and cure in what it cannot know. At this point the concept of sage-knowledge or no knowledge is introduced by the Taoists. This is really not knowledge in the ordinary sense. Knowledge, as we understand it in the West, involves the selection of a certain event or quality as the object of knowledge. Sage-knowledge does not do so. It concerns an understanding of what the East calls Wu or non-being. The Wu transcends events and qualities; it has no shape, no time.  As a result, it cannot be the object of ordinary knowledge”.21

 A typology of language used in narratives

This paper adopts Ralph Siu’s proposed three kinds of language: ‘descriptive’, ‘action’, and ‘poetic’, saying of the first of these:

“The aim of descriptive language is to record or transfer accurate data and messages. It is used in scientific treatises, financial statements and blueprints. Words are exchanged among individuals with intuitive acquaintance as well as systematic knowledge. They refer to things and events primarily outside of the subjective participants. Disagreements can be resolved by objective observations”.22

Siu’s second category – ‘action’ language – “does not stress exactness. Its purpose is to incite activity, to inhibit participation, or to arouse passions. Action language abounds in advertisements, rituals, commands, and sermons…The eloquence of action language joins the head of the speaker with the heart of the listener. Action is incurred without the obstacle of thought”.23

Siu’s third category – ‘poetic’ language – “induces a mood”:

“We observe its charm in a poem, a symphony, a painting, a ballet. It stresses the quality of emotions rather than the utility of experience…. “The composition of poetic language varies greatly. In poetry the context cannot be divorced from the verse, the melody, and the rhythm. The pleasing euphony and the elliptical words blend with the sensuous beauty of the metered sounds….”23

More generally, Siu observes,

There are certain sensations – the woe of a wail, the serenity of a scene, the humour of a joke, the biting of a sarcasm, the devotion of a liturgy, the ecstasy of a song – that can only be suggested by poetic language. There is no direct statement. Yet what can be more explicit a portrayal of the profundity of human reasonableness than an essay of Montaigne, a sharing of deep sorrow than the Pietà of Michelangelo, a union with nature than a verse of Su Tung-po? Poetic language thus becomes descriptive language of another order.”24

Although not an issue in the vignettes, tales and stories considered in this paper, Siu concedes that the boundaries between the types of language may not necessarily be clear-cut, noting that:

“… descriptive words themselves are tinged with poetic harmonics. Except for mathematical symbols there are few completely descriptive terms. Meanings of words are muted by the conditions of learning and prior use. There are the personal sensibilities and dispositions towards the sounds and meanings, which colour the descriptive sense. Such psychological connotations elicit subconscious reactions and descriptive language is infected with action”.25

Content and sentiment analysis of Stwertka’s vignettes

In any compilation of information, the author makes a choice of topics and themes to be included. Stwertka’s comparatively short vignettes of elements cover all the elements, but he selects for inclusion the properties and features that he considers appropriate for his audience (i.e. readers and, probably but unadmitted, his publisher). In the Western tradition, his choice can be challenged, with reviewers identifying omissions and errors of fact.26 Each of Stwertka’s vignettes generally names the discoverer and the date, the origin of the name, a little about the properties of the element and the compounds he selects as the most important, the uses of the element (generally agricultural, industrial and medical), and whether the element and/or its compounds are poisonous or of concern to the environment.

Although the vignettes are presented in Stwertka’s book in order of increasing atomic number, there is not a consistent order of topics and themes within the vignettes – which, admittedly, could have made reading the book tedious – and the vignettes are of variable length. More recently discovered elements have not necessarily been described to the same extent as earlier ones (or may have been more difficult to investigate because of their radioactive or other chemical characteristics). Alternatively, for those elements that have been known about for a longer time, their uses or hazards are likely to be better understood – and there may simply be more to write about. Ultimately though, the length of the vignette is likely to be most dependent on Stwertka’s preferences and choices. 

The content in Stwertka’s vignettes covers, to a variable extent, both the ‘good’ and the ‘bad’ aspects of the element being written about. As an example of the ‘good’, he writes a short paragraph in his vignette on gold, to which the underlining of positive emotive words has been added:

“Gold’s resistance to corrosion and its ability to reflect infra-red radiation and prevent excess heating make it an excellent coating for space vehicles. Modern hotels with large windows use glass coated with a thin film of gold to counter the greenhouse effect and prevent their lobbies from becoming overheated. Dentistry and electronics are two other areas in which the inertness of gold is of great value. Gold teeth can last for decades, and gold-coated switches and connectors remain efficient after years of electrical sparking, which always accompanies the opening and closing of switches.27

In contrast, the ‘bad’ is signalled in a paragraph in his vignette on carbon, to which the underlining of negative emotive words has been added:

“… Many dire predictions have been made about the catastrophic consequences of allowing the buildup of carbon dioxide in the atmosphere to continue. Some environmentalists fear that the melting of icecaps and glaciers, from the warmth produced by the gas, will cause the level of the oceans to rise and flood coastal areas. It is also feared that vast global change in climate from the warming process could produce deserts in areas now fertile. A worldwide alert is in effect….”.28

The vignettes are, thus, candidates for sentiment analysis, in which judgements can be made on whether words evoke positive feelings (e.g. “successful,” “excited”, etc.) or negative feelings (e.g. “embarrassed”, “angry”, “disappointed”, etc.). In general terms, emotions of happiness or satisfaction can be considered a positive sentiment. Conversely, annoyance or anger are negative sentiments. If a given circumstance has no effect, then the sentiment is neutral. Table 1 gives the results obtained in the application of sentiment analysis to very simple and short pieces of text.

While the examples of Table 1 might appear simple initially, there is no absolute symmetry of opposite sentiments: ‘I love you’ and ‘I hate you’ do not have the same sentiment percentage of their respective positive and negative polarities. Moreover, the last two lines of the table show that the order of words can also change the overall sentiment of a text composed of several sentences. This observation confirms that sentiment analysis of a group of sentences is an overall assessment – a ‘gestalt’, if you will. The minimum number of words in a piece of text that need to be analysed in order to obtain a valid measure of sentiment has sometimes been raised as a concern, with one commentator indicating that 200 to 300 words would usually suffice.29 This concern is more appropriate in marketing research in which short texts from social media messages or responses to questionnaires are used rather than in longer narratives such as those analysed in this paper.

Various versions of sentiment analysis have been developed for use with social media as well as for evaluating products and services, of which one of the simplest is Sentiment Analysis: A Definitive Guide.30  In this approach, the sentiment is expressed in terms of ‘polarity’ (negative, neutral, or positive), with the confidence shown as a percentage. This technique can also be applied to narrative text. From this polarity and percentage, a ‘sentiment score’ can be defined in which negative sentiment can have scores ranging from 0 to 0.99, neutral sentiment can have scores ranging from 1.0 to 1.99, and positive sentiment can have scores ranging from 2.00 to 3.00, as illustrated in Fig. 1.

The Sentiment Analysis: a Definitive Guide’s website’s ‘generic’ version of sentiment analysis was applied to the complete text of each of Stwertka’s descriptive (in Siu’s typology) vignettes, the result of which is shown in Fig. 2. It might have been expected that descriptive text of this type would be of neutral sentiment (i.e. a sentiment score of between 1.00 and 2.00). However, for most elements, the sentiment is positive (i.e. sentiment scores between 2.0 and 3.0). Notable exceptions – all of which have negative sentiment polarity – are the halogens of the first three rows of the periodic table (i.e. fluorine, 0.137; chlorine, 0.142; bromine, 0.092) and two of the noble gases: neon (with a score of 0.119) and argon (with a score of 0.073). Caution should be exercised with the sentiment scores of the ‘vignettes’ of the elements in the seventh row of the periodic table (i.e. Fr – Og), because there was sparse chemical information at the time Stwertka was writing about them, attributable to their comparatively recent discoveries and/or the difficulties of preparation of sufficient of the element for analysis.

The haiku as a ‘microvignette’ – and its sentiment analysis

In English versions the Japanese haiku generally comprises three unrhymed lines of five, seven, and five syllables.31 A haiku often features an image, or a pair of images, meant to depict the essence of a specific moment in time, and is here considered an example of Siu’s ‘poetic’ language. The haiku may be used to briefly portray facts, but it may also express opinion. Facts, opinions and combinations of both all feature in a recently published collection of haiku related to the chemical elements,32 examples from which are given below:

  • Fact: for krypton (Kr) – “For twenty-three years nobly accepted the quest to measure metres”; for astatine (At) – “Naturally scarce. Less than an ounce to be had in the whole of the earth’s crust”;
  • Opinion: for manganese (Mn) – “Avoid confusion. In place of your own name, write “NOT magnesium”; for niobium (Nb) – “Are you still pining for the name columbium, its lost poetry?”;
  • Combination: for nihonium (Nh) – “A stable island? Might heavier isotopes linger for longer?”; for polonium (Po) – “Hidden in pitchblende, gleaned by Marie and Pierre, their radiant child”.

Because of this variation in form of the haiku – encompassing fact, opinion, or a combination thereof - their sentiment scores might be expected to vary widely throughout the periodic table, as is apparent from Fig. 3. Many of the haiku have neutral sentiment scores (i.e. between 1.0 and 2.0) and although most elements have lower sentiment scores in their haiku than in their vignettes, there is also no obvious concordance between the sentiment scores of Soon Lee’s haiku and Stwertka’s vignettes.

Application to narratives

Inevitably, the process of allocating words to emotions and then counting them either for chapters or other defined stages in a story has become computerised,33 after which the ‘highs’ and ‘lows’ can then be manipulated to generate a simplified graph of sentiment versus the timeline of a story,34 as shown in Fig. 4. In fact, the shape of the plot of sentiment with the course of the story often varies with Christopher Booker’s stages in the various ‘types’ of fiction he proposes.35 This was demonstrated in a sentiment analysis of The Double Helix as the story unfolds of the discovery of the structure of DNA36 from the perspective of one of its discoverers,37 and in a similar sentiment analysis of the progression of many of Primo Levi’s experience-based stories about the periodic table.38 The same idea has also been applied to historical narratives related to the periodic table,39 to which this paper could be considered a sequel.

Fig. 4. Variation of sentiment with ‘proportion of story’, effectively ‘time’. The curve can be smoothed to reduce the intensity of mood swings, either by using computer packages or by calculating moving averages.

Action narratives – I: Sam Kean’s ‘tales’ of the periodic table

Each of Stwertka’s vignettes focused on a particular element. By comparison, Kean’s tales in his award-winning (attaining second place in 2010’s Royal Society of London's book of the year) The Disappearing Spoon and Other True Tales (and its sequel targeted at younger readers) address the characteristics of chemical elements within a wider theme. This contextualisation is clear from the table of contents and the chemical elements included in each chapter (Table 2 and Table 3), albeit complicated by some of the chapters featuring more than one element. A review which describes Kean’s books of tales admirably observes:

“The beauty of the periodic table is that its structure and organization predicts all manner of important recurring chemical properties: electronegativity, atomic radius, electron affinity, ionization energy, metallic or nonmetallic character, valence, and so on. The transcendence of this framework is usually lost on beginning chemistry students who struggle to grasp the deeper significance of the table. Kean’s book will be of no help to them, as its focus is on the stories rather than the intricacies and implications of the table itself…. stories are what bring science to life, and Kean tells his yarns magnificently.”40

Action narratives – II: Primo Levi’s ‘stories’

Sometimes referred to as the ‘other’ periodic table, Primo Levis’s collection of stories bearing the title The Periodic Table40 (originally published in Italian as Il Sistema Periodico in 1975), was ranked by the Royal Institution of Great Britain in 2006 as the best science book ever published.42 Acknowledging the 100th anniversary of Levi’s birth on 31 July 1919, London-based science writer Philip Ball noted:

“He [Levi] opened up science, and chemistry in particular, to an audience that would have never otherwise given it a thought. He celebrated the culture that was shared, he said, by ‘Empedocles, Dante, Leonardo, Galileo, Descartes, Goethe and Einstein’, as well as by ‘the good craftsmen of today, or the physicists hesitating on the brink of the unknowable’.” 43

Although Levi’s 21 stories relate to his education and employment in chemistry, with each story carrying the name of a chemical element, they are also flavoured by his personal relationships and life experiences (Table 4) to a greater extent than is apparent for the scientists mentioned in Kean’s ‘tales’.

Comparison of sentiment analysis for Stwertka’s vignettes, Kean’s tales and Levi’s stories

Each of Kean’s ‘tales’ and Primo Levi’s stories comprises a few thousand words, for which the complete text of the tale or story was used to determine the sentiment polarities, percentages and scores (as depicted in Fig. 1).  The results of these sentiment analyses are listed for the elements that Primo Levi used in his stories, along with those results determined for Stwertka’s vignettes, using the same procedure previously outlined, in Table 5 and compared in Fig 5.

Although the order of vignettes in Stwertka’s book is strictly that of the elements’ atomic numbers, Kean and Levi were free to choose the order in which their respective tales and stories are presented. Fig. 5A and Fig. 5B indicate that in each of Kean’s books of tales of those elements that also feature in Levi’s book there is a maximum in the variation of sentiment, but not in the same place. For the standard version of the book the maximum occurs in chapter 5 (involving the element nitrogen), while for the young person’s version the maximum occurs at chapter 10 (involving the element silver). Kean’s standard version assigns elements to chapters, but there is no obvious rationale for the order of chapters. Perhaps the order of elements in the chapters in these books was simply an editorial decision based on inferred readability of the various chapters; this could explain the sentiment-maximum in the plots as an attempt to revive any flagging interest in a person reading the book from cover to cover, and might also justify the different order of chapters in the two versions of the book.

In contrast, the order of the chapters in Levi’s book is straightforward: it is essentially chronological. The variation of sentiment across the stories (Fig. 5C) has a minimum about half-way through the book. This corresponds to the ‘central crisis’ (e.g. in rags-to-riches stories) or ‘ordeals to be overcome’ (e.g. in stories involving quests) which occurs in many of the types of fictional stories discussed by Booker (2004).35 That the same features occur in fiction and fact-based narratives is reminiscent of Oscar Wilde’s celebrated comment that “Life imitates art far more than art imitates life”.44

Although highly variable, most of the sentiment for Stwertka’s vignettes and for Kean’s tales are of positive polarity, in contrast to Levi’s stories, which are dominantly of negative polarity. Because the sentiment of the overall narrative is a combination of strands – one related to the chemical element, the other to the human or social aspect of discovery, experiment, or experience – such a variation is to be expected.

It might also be expected that the sentiment scores would be associated with trends in chemical reactivity in the periodic table (e.g. ionisation energy45 and electronegativity46) for Kean’s tales, Levi’s stories and Soon Lee’s haiku, and this association is shown in Fig. 6. Similar trends are expected between scores and other measures of chemistry, in particular the price of chemical elements (indicative of the combination of reactions in industrial extraction and purification processes)47 and atomic potential (the ratio of ionic charge to ionic radius,48 a parameter considered as indicative of natural systems and processes).49 These trends are shown in Fig. 7. Correlation coefficients for linear regression between sentiment scores and any of these parameters are very low, as shown in Table 6, but are markedly higher if the data are considered as fitting a polynomial regression (Table 7), an example of which is shown in Fig. 8. Table 8 shows the same data as Table 7, but is shaded to facilitate the comparison of the representatives of Siu’s (1957) typology of narratives.

Table 8 shows that the sentiments of Stwerka’s descriptive texts have the strongest association with ionisation energy, electronegativity and ionic potential, while the sentiments of Kean’s action narratives have strong associations with the price of the chemical elements. The sentiments of Levi’s action narratives (which relate to his own employment as a chemist) have a modest association with ionisation energy, electronegativity, and – particularly – ionic potential, while the sentiments of Soon Lee’s haiku are most strongly associated with electronegativity. The trends for the five books analysed in this paper in respect of chemical fingerprints are shown in Fig. 9.  As expected, the trends for the two versions of Kean’s book are similar. The trend for Levi’s book is muted, but resembles Kean’s, while the trends for the descriptive narratives of Stwertka and the haiku of Soon Lee are quite different.

Fig. 9. Correlation coefficients for polynomial regression versus chemical fingerprint for types of narrative: descriptive – Stwertka; action – Kean and Levi; poetic – Soon Lee

A digression on narratives of the same element: a comparison of Stwertka’s and Levi’s ‘hydrogen’ 

Stwertka’s collection of vignettes and Levi’s collection of stories both include a narrative related to the explosion of hydrogen. An explosion involving hydrogen can be potentially dangerous (as well as exciting), and therefore inferred to be of negative sentiment, as in Levi’s story. Of course, the explosion can be exciting (as well as potentially dangerous) and therefore perceived to be of more positive sentiment, as inferred for Stwertka’s narrative relating to hydrogen’s discovery and Henry Cavendish’s (1776) experiment (Fig. 10).50 Primo Levi was less fortunate than Cavendish in his electrolysis of salty water: he and his student friend Enrico collected the two gases produced (i.e. hydrogen and oxygen) together in a closed container, a similar version of which is available online (Explosive water through electrolysis) together with an associated video (Make Fuel from Water),51 rather than an open container as Cavendish had done. The container was destroyed in the ensuing explosion, which Levi described thus: “There was a small explosion, small but sharp and angry, the jar burst into splinters”.52 Although the explosion of hydrogen features earlier in Stwertka’s vignette than in Levi’s story, the overall shape of the sentiment plot shown in Fig. 11 is similar for both narratives.

This result indicates that similar events – or even the same event – can result in very different narratives with very different sentiments both as the story progresses and overall, especially if the ‘observers’ are different. Although the modern Western scientific mind might challenge the scientific legitimacy of these different interpretations of ‘facts’, much older Eastern mysticism would probably find them acceptable, essentially as Edward Said has explained:
“A book on how to handle a fierce lion might… cause a series of books to be produced on such subjects as the fierceness of lions, the origins of fierceness, and so forth. Similarly, as the focus of the text centers more narrowly on the subject – no longer lions but their fierceness – we might expect that the ways by which it is recommended that a lion’s fierceness be handled  will actually increase its fierceness, force it to be fierce since that is what it is, and that is what in essence we know or can only know about it.
“A text purporting to contain knowledge about something actual, and arising out of circumstances similar to the ones I have described, is not easily dismissed. Expertise is attributed to it… Most important, such texts can create not only knowledge but also the very reality they appear to describe.”53

Conclusion

This paper highlights the range of results of sentiment analysis obtained in its application to narratives related to the chemical elements of the periodic table. This range incudes different results for sentiment analysis of narratives considered to be essentially descriptive of facts (e.g. Stwertka’s vignettes), narratives which include actions around either the discovery of those facts (e.g. Kean’s tales) or actions related to experiences of a chemist (e.g. Levi’s stories) and narratives with a more poetic or creative character, exemplified here by Soon Lee’s haiku. This suggests a continuum can be envisaged between Western pragmatism on the one hand – most obviously shown in descriptive narratives – and Eastern mysticism on the other – most obviously shown in poetic works, with action narratives occupying a position between or even overlapping with these end-points.

Such a continuum suggests that Western and Eastern thought may not necessarily be as independent from each other as has been often portrayed.54 Indeed, it has been suggested recently that the sustained research undertaken by Joseph Needham and reported in his multi-volume Science and Civilisation in China (1954-2004)55 “enriches philosophy as a whole by avoiding the idiosyncrasies inherent to conceiving the world from the point of view of one culture or geo-social context”.56 Such an inter-cultural link is implicit in popular histories,57 but is explicit in Fernández-Armesto’s statement:

“For science to thrive, the idea of it was not enough. People needed to observe nature systematically, test the ensuing hypotheses, and classify the resulting data. The method we call empiricism answered those needs. Where did it come from? We can detect intellectual origins in Taoist doctrines of nature, and early applications in medicine.”58

References and notes

  1. Rayner-Canham, G. Organizing the transition metals. In: Scerri, E., Restrepo, G. (eds): Mendeleev to Oganesson: A Multidisciplinary Perspective on the Periodic Table. Oxford University Press: New York, 2018, pp. 195-205; Scerri, E.R.; Parsons, W.F. What elements belong to Group 3 of the periodic table? In: Scerri, E., Restrepo, G. (eds), Mendeleev to Oganesson: A Multidisciplinary Perspective on the Periodic Table. Oxford University Press: New York, 2018, pp. 140-151.
  2. Railsback, L.B. The earth scientist’s periodic table of the elements and their ions: A new periodic table founded on non-traditional concepts. In: Scerri, E.; Restrepo, G. (eds): Mendeleev to Oganesson: A Multidisciplinary Perspective on the Periodic Table. Oxford University Press: New York, 2018, pp. 206-218.
  3. Scerri, E. The Periodic Table Its Story and Its Significance, 2nd edition. Oxford University Press: New York, 2020, p. 381, fig. 13.5; Seaborg, G. Plutonium: The ornery element. Chemistry, 1964, 37 (6, June), 12–17 [Benfey’s representation of the periodic table is on p. 14].
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  5. Scerri, E. 2020, op. cit., p. 381, fig. 13.7; Scerri, E. The past and future of the periodic table. American Journal of Science, 2008, 96, 52-58 [Stewart’s galaxy representation is on p. 58].
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  17. An example is: Hipkins, R.; Arcus, C. Teaching science in context: challenges and choices. In: Bell, B., Baker, R. (eds): Developing the Science Curriculum in Aotearoa New Zealand. Longman: Auckland, 1997, pp. 110-113.
  18. An example is: Matthews, M.R. Challenging NZ [New Zealand] Science Education. Dunmore Press: Palmerston North, 1995.
  19. Gash, H. Constructivism and mysticism. In: Lasker, G.E.; Hiwaki. K. (eds.) Personal and Spiritual Development in the World of Cultural Diversity, Vol XV. International Institute for Advanced Studies: Tecumseh (Ontario), 2018, pp. 7-11.
  20. Hood Jr., R. W. Constructivism and mysticism: Beyond epistemological constraints. Constructivist Foundations 2019, 15 (1), 12–13, https://constructivist.info/15/1/012.
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  23. Siu, 1957, op. cit. p. 48.
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