Elements, Letters and Numbers

There are 14 elements represented by a single letter on the periodic table: Hydrogen (H), Boron (B), Carbon (C), Nitrogen (N), Oxygen (O), Fluorine (F), Potassium (K), Yttirium (Y), Iodine (I), Tungsten (W), Uranium (U), Sulfur (S), Phosphorus (P), and Vanadium (V). There atomic numbers representing the number of protons in their nuclei are as follows:

• Hydrogen (H) - 1
• Boron (B) - 5
• Carbon (C) - 6
• Nitrogen (N) - 7
• Oxygen (O) - 8
• Fluorine (F) - 9
• Potassium (K) - 19
• Yttrium (Y) - 39
• Iodine (I) - 53
• Tungsten (W) - 74
• Uranium (U) - 92
• Sulfur (S) - 16
• Phosphorus (P) - 15
• Vanadium (V) - 23

An interesting question is how many English words can be formed from this mixture of vowels and consonants. Let's repeat the letters below and sort them into two categories:

I, O, U, B, C, F, H, K, N, P, S, V, W, Y

Here are some words that ChatGPT found that can be formed using these letters. It's not an exhaustive list.

hub, cub, sin, pin, win, his, if, vow, soy, spy, wok, sky, sun, pun, fun, kin, ink, no, on, up, us, so, is, by, icy, coy, buy, boy, fob, hob, sob, bow, cow, how, low, now, own, won, you, ivy, ups, cup, pus, sup, pin, nip, hip, sip, cop, sop, vow, sow, soy, cosy, coif, sync, piny, fowl, foul, busy, busybody, snow, snowy, cow, cowpox, cup, cupful, cusp, sushi

Each of these words can be assigned a numeric value on the basis of the atomic numbers associated with their letters. For example, hub = 1 +  92 + 5 = 98 and sushi = 19 + 92 + 19 + 1 + 53 = 184. What I've created is a gematria of sorts.

Of course, one does not need to limit oneself just to elements with a single letter. An article has been written titled List of Words Made From Periodic Table Element Symbols. It was posted on June 17, 2014 by Anne Helmenstine  and updated on January 9, 2023. Figure 1 shows a screenshot displaying the words PHYSiCs, BaNaNa, FUN and GeNiUS.

Figure 1: 

Using the atomic numbers again. It can be seen that PHYSiCs = 124, BaNaNa = 88 and GeNiUS = 168. The words in the article are listed alphabetically and in camel case. Figure 2 shows a screenshot of the C words.

Figure 2

The article only considers the words that can be made from the chemical elements and does not assign the words any numerical values, based on atomic numbers or anything else. That was my idea which is why I'm posting to this mathematics blog and not my Pedogogical Posturing blog. 

I can't think of any practical use for this gematria at the moment but that's of no consequence. Perhaps something will occur to me later. Like in standard Greek or Hebrew gematria, the atomic number assignments to elements will create links between words that add to the same sum. Figure 3 shows a table of elements and atomic numbers, sorted by element.

Figure 3: source

Figure 4 shows a periodic style table that shows the elements sorted by atomic number.

Figure 4: source

Of course there are many words that cannot be formed by combining the letters together e.g. light. We can start with Li (Lithium) but after that no further progress can be made. An interesting SageMath project would be to create a dictionary of elements and their atomic numbers and use this to automatically output the number of any word that was input.

ChatGPT created a list of triplets containing chemical name, chemical symbol and atomic number. Here it is:

[("Hydrogen", "H", 1), ("Helium", "He", 2), ("Lithium", "Li", 3), ("Beryllium", "Be", 4), ("Boron", "B", 5), ("Carbon", "C", 6), ("Nitrogen", "N", 7), ("Oxygen", "O", 8), ("Fluorine", "F", 9), ("Neon", "Ne", 10), ("Sodium", "Na", 11), ("Magnesium", "Mg", 12), ("Aluminum", "Al", 13), ("Silicon", "Si", 14), ("Phosphorus", "P", 15), ("Sulfur", "S", 16), ("Chlorine", "Cl", 17), ("Argon", "Ar", 18), ("Potassium", "K", 19), ("Calcium", "Ca", 20), ("Scandium", "Sc", 21), ("Titanium", "Ti", 22), ("Vanadium", "V", 23), ("Chromium", "Cr", 24), ("Manganese", "Mn", 25), ("Iron", "Fe", 26), ("Cobalt", "Co", 27), ("Nickel", "Ni", 28), ("Copper", "Cu", 29), ("Zinc", "Zn", 30), ("Gallium", "Ga", 31), ("Germanium", "Ge", 32), ("Arsenic", "As", 33), ("Selenium", "Se", 34), ("Bromine", "Br", 35), ("Krypton", "Kr", 36), ("Rubidium", "Rb", 37), ("Strontium", "Sr", 38), ("Yttrium", "Y", 39), ("Zirconium", "Zr", 40), ("Niobium", "Nb", 41), ("Molybdenum", "Mo", 42), ("Technetium", "Tc", 43), ("Ruthenium", "Ru", 44), ("Rhodium", "Rh", 45), ("Palladium", "Pd", 46), ("Silver", "Ag", 47), ("Cadmium", "Cd", 48), ("Indium", "In", 49), ("Tin", "Sn", 50), ("Antimony", "Sb", 51), ("Tellurium", "Te", 52), ("Iodine", "I", 53), ("Xenon", "Xe", 54), ("Cesium", "Cs", 55), ("Barium", "Ba", 56), ("Lanthanum", "La", 57), ("Cerium", "Ce", 58), ("Praseodymium", "Pr", 59), ("Neodymium", "Nd", 60), ("Promethium", "Pm", 61), ("Samarium", "Sm", 62), ("Europium", "Eu", 63), ("Gadolinium", "Gd", 64), ("Terbium", "Tb", 65), ("Dysprosium", "Dy", 66), ("Holmium", "Ho", 67), ("Erbium", "Er", 68), ("Thulium", "Tm", 69), ("Ytterbium", "Yb", 70), ("Lutetium", "Lu", 71), ("Hafnium", "Hf", 72), ("Tantalum", "Ta", 73), ("Tungsten", "W", 74), ("Rhenium", "Re", 75), ("Osmium", "Os", 76), ("Iridium", "Ir", 77), ("Platinum", "Pt", 78), ("Gold", "Au", 79), ("Mercury", "Hg", 80), ("Thallium", "Tl", 81), ("Lead", "Pb", 82), ("Bismuth", "Bi", 83), ("Polonium", "Po", 84), ("Astatine", "At", 85), ("Radon", "Rn", 86), ("Francium", "Fr", 87), ("Radium", "Ra", 88), ("Actinium", "Ac", 89), ("Thorium", "Th", 90), ("Protactinium", "Pa", 91), ("Uranium", "U", 92), ("Neptunium", "Np", 93), ("Plutonium", "Pu", 94), ("Americium", "Am", 95), ("Curium", "Cm", 96), ("Berkelium", "Bk", 97), ("Californium", "Cf", 98), ("Einsteinium", "Es", 99), ("Fermium", "Fm", 100), ("Mendelevium", "Md", 101), ("Nobelium", "No", 102), ("Lawrencium", "Lr", 103), ("Rutherfordium", "Rf", 104), ("Dubnium", "Db", 105), ("Seaborgium", "Sg", 106), ("Bohrium", "Bh", 107), ("Hassium", "Hs", 108), ("Meitnerium", "Mt", 109), ("Darmstadtium", "Ds", 110), ("Roentgenium", "Rg", 111), ("Copernicium", "Cn", 112), ("Nihonium", "Nh", 113), ("Flerovium", "Fl", 114), ("Moscovium", "Mc", 115), ("Livermorium", "Lv", 116), ("Tennessine", "Ts", 117), ("Oganesson", "Og", 118)]

This is suitable for pasting into SageMath and doing further analysis. Here is a permalink to an algorithm that will input chemical symbols that make a word (in camelcase) and output the word in uppercase together with its atomic number value. I've used "B", "O", "Th" and "Er" as input and gotten "BOTHER has an atomic number of 171" as output.

Wordle Statistics

In my Pedagogical Posturing blog, I recently posted about Wordle but in post I want to focus on the statistics of the letter frequencies. Of primary interest of course is how frequently the various letters of the alphabet occur within the Wordle "universe". In A Mathematician's Guide to Wordle, it's explained that:

Wordle has both a ‘source’ word list and a ‘target’ word list. You can guess anything from the source word list (which is the 5 letter words from CSW19) ... The target word list is a small hand-curated subset of less than 2500 of these words.

CSW stands for Collins Scrabble Words that consists of 279,496 words. The article just quoted from goes on to say that:

It is well known that when you order letters by frequency of appearance in English words, you get ETAOIN SHRDLU. However ... if you take all 5 letter words in CSW19, you get SEAORY LTNUDY.

Figure 1 shows the full sequence of letters and their relative frequencies. The commentary on this chart includes:

• In fact, they're essentially tied, S appearing only three more times than E in the 64,860 letters in the word list. That's a 0.0046 percentage point difference.
  
  
• After those two, the distribution doesn't fall off that much. A is 9.2%, O is 6.8%, R is 6.4%.
  
  
• The top 10 most frequent letters in the list make up 67% of the occurrences. That's all five vowels, plus the Wheel of Fortune consonants, R S T L N.
  
  
• For those of you wondering about that "other" vowel, Y is 12th at 3.2%.
  
  
• The least common 5 letters, Q X J Z V, combine for a mere 2.8% of the occurrences, about the same as the letter H, the 16th ranked letter.
  
  
• The most common letter, S, is overwhelmingly more likely to be in fifth position (59%). Otherwise, it's more likely that it appears first (23%) than in any of the middle positions combined.
  
  
• Vowels occur most commonly in the second position. That is unless you're E, which is more common in the fourth spot (35%) than second (24%). Again, "-ed" words are a plausible explanation.
  
  
• For you Y fans: The pseudo-vowel shows up in the last spot 63% of the time when it appears.
  
  
• Of the most common letters, L has the most level distribution, appearing most in the third position (25%) and least in the last position (14%).
  
  
• Words in which letters appear twice make up a little more than 35.8% of the accepted word list.
  

Figure 1: source

Looking at Figure 1, it would seem that the AROSE would be a good starting point. Figure 2 shows the positions within a word of the most common letters.

Figure 2: source

Figure 2 shows that S occurs most frequently in the final position and so the previous choice of AROSE is perhaps not as good a choice as it first seemed. There are no permutations of this word that have S in the final position. Finally Figure 3 shows the frequency of letters occurring twice in a word compared with their overall frequency.

Figure 3: source

Given that both E are S more likely to occur twice in a word, it would seem that EASES might be good choice of starting word. Some other suggested starting words are:

• TARES
• SORES
• CARES
• CORES
• REAIS
• BLAHS
• SOLAR

There are a lot of factors to consider but this post is at least a start and will definitely improve my Wordle prowess.