# Talk:Tetrahedral number

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'Italic text(cur) (last) . . 00:49, 16 Apr 2004 . . 66.32.65.129 (Corrected link; please do not interpret as vandalism)

66.32.65.129, please explain: why you think anyone might interpret your edit as vandalism? Your edit appears to me to be a simple correction changing a link from a disambiguation page to a link to a specific page. PrimeFan 13:21, 16 Apr 2004 (UTC)

## Animated Image

Is there WP policy on animated images? Normally I'd think a still would be more appropriate. --Aioth 02:37, 19 August 2006 (UTC)

Then you would see a 2D image, but its a 3D object.

One question I would like, might anyone know how to derive a tetrahedral number?ZakTek 13:59, 1 January 2007 (UTC)

## Tetrahedral root?

I see that there is a test for triangle numbers in the Triangle number article. It's what I would call a triangle root. It is a great way to test for triangle numbers. Now, do you have a similar formula for testing tetrahedral numbers? This would be a tetrahedral root. (Up to the challenge or did I just stump you)? :)

FYI: I am the moderator for [ http://forums.delphiforums.com/figurate/start]

The test to check for valid triangle numbers is just the triangular number generating formula, $T_{n}={\frac {n(n+1)}{2}}$ , solved for n. It involves finding the roots of a quadratic. Notice the original formula, when put in standard form, will have n2 in it.
On the other hand, the tetrahedral number generating formula, $T_{n}={\frac {n(n+1)(n+2)}{6}}$ , will result in a cubic when put in standard form. So, your question is equivalent to asking, "Is there a simple formula that can find the roots of a cubic?" The answer: no. Is there a guaranteed way to find the roots of a cubic? Answer: Yes. I would not call it simple. -- Wguynes 14:29, 17 June 2007 (UTC)

## Basic information missing

The article gives a few examples of tetrahedral numbers that are also triangular numbers. The obvious question: are these all there are, or if not, how many are there, or are there infinitely many? Surely the answer must be known 91.105.16.28 14:06, 25 June 2007 (UTC)

It's limited, there are only five to be exact. I'll be adding that information soon. The page is also missing one number that's also both triangular and tetrahedral (the number 1). The edit will be saved in a few minutes from now.

70.139.121.174 (talk) 18:36, 31 December 2007 (UTC)

In late 2009, while working on a new model of the periodic table, I realized that every other Pascal tetrahedral number (4,20,56,120....) was also the atomic number of every other alkaline earth (2s2,4s2,6s2,8s2....). Intermediate alkaline earth atomic numbers are the arithmetic means of the Pascal tetrahedral numbers, but the intermediate Pascal numbers are not, but rather have a linearly increasing offset (so alkaline earth 2 (He controversial in this placement but not for electron configuration 1s2) is Pascal 1+1, 12 (3s2)= 10+2, 38 (5s2)=35+3, 88 (7s2)= 84+4 and so on.

Given the general rule against new research (it IS published online), I'd like advice as to whether this is something that might be useful for either the Tetrahedral number or Pascal Triangle pages. It is something that makes one think, and it is true (as any cursory examination of both the Pascal Triangle and the periodic table will show).

More controversial, though, is using this relation to help to justify both the Janet Left Step Periodic Table of 1927-28 (which can be found in the Wiki section on alternative periodic tables) and my new tetrahedral model of the periodic relation (a solid not exactly equivalent to a table).

In any case, please let me know whether at least the diagonal relation is worth noting. In the Janet table the alkaline earths are rightmost, making the Pascal numbers special in this regard. The Janet table is usually dismissed by chemists because they are overawed by electron ionization energies, while going by electron configurations and dimensions of blocks really does justify the Janet system. But again, more controversy...

Lardehartet (talk) 06:35, 11 May 2010 (UTC)

Article talk pages are meant for improving the article, not for speculating on the article's subject matter.—Tetracube (talk) 18:28, 11 May 2010 (UTC)

References link is dead- let someone in authority deal with it. —Preceding unsigned comment added by 71.127.246.197 (talk) 14:59, 14 September 2010 (UTC)

## The Twelve Days of Christmas?

T12 is the total number of gifts sent by the singer's "true love" during the course of the Twelve Days of Christmas. Is this worth including as a light-hearted application of Tetrahedral Numbers? Portnadler (talk) 15:14, 14 December 2010 (UTC)

Isn't that the twelfth triangle number? I didn't think that on day three, for example, they gave the gifts from days one and two /again/. Finbob83 (talk) 19:51, 1 January 2012 (UTC)
Yes they did. See http://www.carols.org.uk/the_twelve_days_of_christmas.htm --Portnadler (talk) 15:22, 20 January 2012 (UTC)

Somebody claimed "This is false" and then another person removed the entire section. I have reinstated it with a reference to an external site that demonstrates its truth. Portnadler (talk) 12:04, 9 September 2015 (UTC)

## Tetrahedral numbers in the Periodic Table

In the Left-Step Periodic Table of Janet (see second illustration at http://en.wikipedia.org/wiki/Alternative_periodic_tables), 'periods' based on quantum filling of orbitals, rather than on the chemical properties of inert gases, end with the alkaline earths. Also, in the Janet table, every period is paired for length. Since the number of elements in each period (whether the traditional or Janet representation) is both half and double square (2,8,18,32,50...), this means that the number of elements in each pair of same length periods is a square.

Note now that every other alkaline earth atomic number is equal to every other tetrahedral number: 4,20,56,120. Intermediate alkaline earth atomic numbers 1,12,38,88 differ from the corresponding tetrahedral numbers by amounts increasing monotonically: 1-0=1, 12-10=2, 38-35=3, 88-84=4, and so on. The two systems share some features, but differ in others.

Tetrahedral numbers are based on the sums of squares, either all even or all odd- the numbers that match those of the alkaline earths are the sums of even squares, those that don't are the sums of odd squares. In the Janet table each period length is half of an even square.

The mismatch between the use of the tetrahedral numbers comes from the construction of a tetrahedron (say of close-packed spheres) out of flat layers that have triangular numbers of forms as in the Pascal Triangle. The Periodic Table operates differently here, but one can construct a model of it using instead stacks of skew rhombi (skewed up to a tetrahedral edge-edge angle). Such rhombi contain square numbers of forms (so dual periods), yet each addition just makes the initial tetrahedron (of 4 forms) bigger. 67.81.236.32 (talk) 05:06, 17 January 2012 (UTC)

## Double summations and tethrahedral numbers

Hello, This is a very short paper about the relation between double summations and tetrahedral numbers. Let me know if it could be interesting for this page: http://vixra.org/pdf/1103.0031v1.pdf

Best regards,Marcokrt (talk) 01:51, 27 January 2012 (UTC)

## Animated image

Animated image says that each row is for the first 5 triangular numbers. Should it not be the first 5 tetrahedral numbers? — Preceding unsigned comment added by RulerofKnowledge (talkcontribs) 21:38, 17 January 2013 (UTC)

No: 1, 3, 6, 10, 15 are the first five triangular numbers. The running total descending from the top, 1, 4, 10, 20, 35 comprises the first five tetrahedral numbers. Jackaroodave (talk) 13:52, 16 August 2021 (UTC)