For reasons not clear to me, probably due to the inertia of history, the Periodic Table of the Chemical Elements seems to have fossilised into just one particular layout in almost all textbooks of Chemistry. There are however many different layouts of the data that have been designed over the years. The "Chemistry Coach" site used to give links to a comprehensive selection of these alternative tables, but has now vanished from the web. However the "Chemogenesis" site is probably even more comprehensive, and is good on precursors to Mendeleev. Wikipedia also has some pages and links on these designs, and others.
Wikipedia: Alternative periodic tables
As may be expected I have my own favoured version of the table, shown below. It is one I devised while still at school, in 1958, when I was 18. Around that time or shortly after I sent it to New Scientist magazine but don't recall hearing any more about it. The designs nearest to my version, quoted by "Chemistry Coach", are those associated with Andis Kaulins (1972) or Albert Tarantola (1975), although the basic idea goes back to Charles Janet (1929) and is often rediscovered. I now note that Valery Tsimmerman (2006) has come up with something very similar, but three-dimensional.
My design, which uses the quantum numbers l and n as coordinates, resulting in a more symmetrical table, was eventually published, in 1988, in my own Games and Puzzles Journal, though by then I had forgotten what the quantum numbers signified. However, thanks to the websites cited here I have now recovered some knowledge. Roughly speaking the quantum number n, which numbers the columns and the periods, is a measure of energy. The quantum number l, which groups the rows, is a measure of angular momentum. It will be seen that at the end of each full period the table up to that point is symmetric about the vertical median. This symmetry is not apparent in the usual representations.
|l = 3||57 La||89 Ac|
|58 Ce||90 Th|
|59 Pr||91 Pa|
|60 Nd||92 U|
|61 Pm||93 Np|
|62 Sm||94 Pu|
|63 Eu||95 Am|
|64 Gd||96 Cm|
|65 Tb||97 Bk|
|66 Dy||98 Cf|
|67 Ho||99 Es|
|68 Er||100 Fm|
|69 Tm||101 Md|
|70 Yb||102 No|
|l = 2||21 Sc||39 Yt||71 Lu||103 Lr|
|22 Ti||40 Zr||72 Hf||104 Rf|
|23 V||41 Nb||73 Ta||105 Db|
|24 Cr||42 Mo||74 W||106 Sg|
|25 Mn||43 Tc||75 Re||107 Bh|
|26 Fe||44 Ru||76 Os||108 Hs|
|27 Co||45 Rh||77 Ir||109 Mt|
|28 Ni||46 Pd||78 Pt||-|
|29 Cu||47 Ag||79 Au||-|
|30 Zn||48 Cd||80 Hg||-|
|l = 1||5 B||13 Al||31 Ga||49 In||81 Tl||-|
|6 C||14 Si||32 Ge||50 Sn||82 Pb||-|
|7 N||15 P||33 As||51 Sb||83 Bi||-|
|8 O||16 S||34 Se||52 Te||84 Po||-|
|9 F||17 Cl||35 Br||53 I||85 At||-|
|10 Ne||18 A||36 Kr||54 Xe||86 Rn||-|
|l = 0||1 H||3 Li||11 Na||19 K||37 Rb||55 Cs||87 Fr||-|
|2 He||4 Be||12 Mg||20 Ca||38 Sr||56 Ba||88 Ra||-|
|n = 1||n = 2||n = 3||n = 4||n = 5||n = 6||n = 7||n = 8|
The colors indicate that the table is formed of 8 periods, of lengths 2, 2, 8, 8, 18, 18, 32, 32, in each of which the elements are arranged downward in sequence of increasing atomic number. The eighth period, shown in white, is incomplete, since the elements with these high numbers of protons become increasingly unstable, and the later ones in the series have only been produced under laboratory conditions and do not occur in nature. These periods are themselves made up of shorter sub-periods of lengths 2, 6, 10, 14. The number of elements in sub-period (n, l) is 4l + 2, which is twice an odd number. The number of elements in period n is thus twice the sum of the first h odd numbers, where h = (n+1)/2 or n/2 according as n is odd or even. The sum of the first h odd numbers, as everyone should know, is the square number h^2, so the length of the full nth period is 2(h^2). Thus the numbers in the sequence 2, 8, 18, 32 are double-square numbers.
The quantum numbers are well explained at these sites:
The table could of course be inverted or rotated. I prefer the orientation shown since, when printed in the form of a wall-chart, the more common and important elements appear in the lower part where a child would read them first. There is however something to be said for the inverted form, since higher values of l indicate that the extra electrons are filling up deeper 'shells' or 'orbits' and the deeper they are the less effect they tend to have on the chemical properties, which is why the Lanthanides (La to Yb) and Actinides (Ac to No) are very similar chemically. Other colour schemes can of course be substituted for the one used here.
(c) 2005, G. P. Jelliss. Revised July 2012 to replace various defunct links. I notice that my design was put on the Chemogenesis list in 2007 but the link there is out of date: Jelliss' Periodic Table.