Pascal,Triangle,and,Tetrahedro education Pascal's Triangle and Pascal's Tetrahedron

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Pascal's tetrahedron or Pascal's pyramid is an extension of the ideas fromPascal's triangle. Many of the properties of Pascal's triangle can be applied(with a little modification) to Pascal's Pyramid. However, in this article, Idiscuss only the direct links between the two, which are even more extensivethan one might initially imagine.I have begun by showing the first 4 layers of Pascal's tetrahedron below:Layer 0:1Layer 1:11 1Layer 2:12 21 2 1Layer 3:13 33 6 31 3 3 1Layer 4:14 46 12 64 12 12 41 4 6 4 1In layer 3, the final row of the layer is 1,3,3,1, row three of Pascal'striangle, the final row of layer 4 the 4th, and so on for all the layers listedabove. In fact, if you write out each of the layers shown above in a centeredequilateral triangle, you will notice that every edge of each triangular layeris that layer's corresponding row in Pascal's triangle.This pattern does, in fact, always continue. Without delving too deeply intorigorous mathematical proofs, you should if you think about it be able to seethat on the edges of layers, you only ever add together two numbers from thelayer above, and so really it's just like Pascal's triangle (where you add twonumbers from the row above) repeated three times at various angles.However, the links run even deeper than this (in more ways than one). Itsnot just about the edges of the pyramid, there are links deep down inside thevery core of the tetraheron.To understand this, we are going to use layer 4 as an example, but thistime, we will not look at an edge but as some of the rows running through themiddle of the layer. For example, is there anything interesting about thesecond to last row, which goes 4,12,12,4? In fact there is (otherwise Iwouldn't be asking). For the moment, let's just forget about the numbersthemselves, and think only about the ratio between them. This gives a1:3:3:1 ratio, as the middle two numbers are thrice the outside two numbers.This ratio happens to be the third row of Pascal's triangle. Is that just acoincidence?Next, let's investigate the third last row in layer 4, which goes 6,12,6.This time, the ratio is 1:2:1, the second row. Something is definitely going onhere. Below is the whole of layer 4, split into rows, with their ratios andwhere they can be found in Pascal's triangle:Layer 4:14 4 - ratio 1:1 (row 1)6 12 6 - ratio 1:2:1 (row 2)4 12 12 4 - ratio 1:3:3:1 (row 3)1 4 6 4 1 - ratio 1:4:6:4:1 (row 4)So, amazingly, every single row in layer 4 is in the ratio of the row inPascal's triangle which has the same number of numbers in it! However, justwhen you thought it couldn't get any more exciting, look at what we have tomultiply the ratios by to get the actual numbers in Pascal's tetrahedron again:1 (1) x 14 4 - (1,1) x 46 12 6 - (1,2,1) x 64 12 12 4 - (1,3,3,1) x 41 4 6 4 1 - (1,4,6,4,1) x 1They are the 4th row of Pascal's triangle! Only now do we truly see theextent of the links between these two patterns of numbers. Every single numberin the pyramid is simply two numbers from Pascal's triangle multipliedtogether. Not only is this in my opinion a beautiful discovery which is anexcellent demonstration of the interconnected nature of mathematics, it makeswhat seemed like the much more complex Pascal's tetrahedron easy to work with.In fact, it is these links that have helped mathematicians to modify theformula for Pascal's triangle to one which applies to Pascal's tetrahedron andeven to suit higher dimensions, so it is certainly a very powerful discovery! Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;} Article Tags: Pascal's Triangle, Pascal's Tetrahedron, Numbers From