Why do triangles in multiple dimensions use pascals triangle

Question

If we take a triangle in 0 dimensions, and we count the number of points it has, we see it only has 1 (since anything in 0 dimensions is a point) |Points| |:----:| |1| If we take a triangle in 1 dimension, we get a line segment, which is made by a side, and 2 points |Lines|Points| |:---:|:----:| | |1| |1|2| A triangle in 2 dimensions is a regular triangle, with 1 face, 3 lines, and 3 points. |Faces|Lines|Points| |:---:|:---:|:----:| | | |1| | |1|2| |1|3|3| In three dimensions, a tetrahedron there are 4 points, 6 lines, 4 faces, and 1 polyhedron |Polyhedrons|Faces|Lines|Points| |:-:|:-:|:-:|:-:| | | | |1| | | |1|2| | |1|3|3| |1|4|6|4|

This pattern coincides exactly with pascal's triangle (without the last '1' at the end of every row). Here, the pascal triangles row coincides with the dimension + 1, and the number in the row is for a certain property present in only some dimensions. Does this pattern continue in higher dimensions, and is there a reason for this?

Answer 1

The underlying reason is that, increasing by one dimension, you add:

• 1 new vertex (dim =0);
• so many edges(dim =1) as previous vertices (dim =0);
• so many 2-facets as previous 1-elements;
  etc.