Properties,Pascals,Pyramid,Pas education Properties of Pascals Pyramid
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Pascal's Pyramid, or Pascal's tetrahedron is an interesting extension of theideas from Pascal's triangle. In this article, I take some of the basicproperties of Pascal's triangle, such as sums of rows and see if they can bemodified to apply to Pascal's tetrahedron.One famous property of Pascal's triangle is that the sums of the rows arethe doubling numbers. Rather than looking at the sums of rows in Pascal'spyramid, we can see if we get any similar patterns when we look at the sums oflayers. This has been done for layers 0 to 4 below:Layer 0:1Total = 1Layer 1:11 1Total = 1 + 1 + 1 = 3Layer 2:12 21 2 1Total = 1 + 2 + 2 + 1 + 2 + 1 = 9Layer 3:13 33 6 31 3 3 1Total = 1 + 3 + 3 + 3 + 6 + 3 + 1 + 3 + 3 + 1 = 27Layer 4:14 46 12 64 12 12 41 4 6 4 1Total = 1 + 4 + 4 + 6 + 12 + 6 + 4 + 12 + 12 + 4 + 1 + 4 + 6 + 4 + 1 = 81The sums of the layers triple each time, producing a formula for the sum ofthe nth layer of 3^n. In fact, this property can tell us something else aboutPascal's pyramid. Things can easily get very complicated with this, so I'm notgoing to try too explain this too much. If we look at the average of thenumbers in each row of Pascal's triangle, we get the following results for thefirst few rows:1, 1, 1.33, 2, 3.2, 5.33, 9.14...Now, if we write down what you have to multiply each term by to get to next,you get1, 1.33, 1.5, 1.6, 1.67, 1.71, 1.75...If you kept on going, you would get a value closer and closer to 2, so youwould get the average of the numbers in the rows eventually doubling each time.If you try this with the average of the layers in Pascal's tetrahedron, youshould find you get a sequence which gets closer and closer to tripling eachtime. This explains why the numbers get large so much faster in Pascal'stetrahedron.We can also look at the symmetry of Pascal's tetrahedron. If you arrangeeach layer as an equilateral triangle, it has rotational symmetry of order 3,and reflective symmetry from each of its corners to the midpoint of theopposite side. This may sound complicated, but let's think about Pascal'striangle for a moment. in each row, every number appears twice unless it is inthe very centre of the row. This is due to the symmetry through the centre ofthe triangle. Pascal's tetrahedron, however, due to slightly more complicatedsymmetry, has every number repeated three times is each layer, with theexception of a number which is sometimes found in the very centre of thetriangular layer, such as the 6 in layer 3.It is clear, therefore, that many of the properties of Pascal's triangleapply in some way to Pascal's pyramid as well, but it is interesting to thinkabout how these patterns have evolved to suit Pascal's tetrahedron, and thereasons for these changes. 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 Pyramid, Pascal's Tetrahedron, Pascal's Triangle, Each Time
Properties,Pascals,Pyramid,Pas