Number Combinations
In Pascal’s triangle, the second set of diagonal numbers running along each edge forms the natural numbers: 1, 2, 3, 4, 5, … Natural numbers are used to count tangible objects – screws, nuts, physical quantities one can hold.
The third set of diagonals along the edges forms the triangular numbers: 1, 3, 6, 10, 15, … Triangular numbers describe something that cannot be touched. They are like the shadow of calculation behind observable phenomena – an abstract structure that does not appear directly, yet governs form and growth.
Proportionality calculation is based in part on the coefficient 1.1(2). Pascal demonstrated through his triangle that numerical combinations can be determined systematically. If we wish to select two objects from a set of six, how many distinct choices exist? By taking the second diagonal of Pascal’s triangle (the natural numbers 1, 2, 3…) and moving one step to the right from the number 6, we obtain 15. This represents the total number of possible selections.
The principle resembles the interplay of light and shadow in a photograph. The six objects correspond to the illuminated area of the image – the visible, tangible part. In the shadow lies the number of possible combinations: not directly seen, yet not hidden either. Pascal’s triangle “exposes” this invisible structure and brings the number 15 into view.
The objects themselves are concrete, but the number of combinations they form is abstract. These two levels are connected: the tangible provides the starting point, while the intangible – the structure of combinations – is the domain of proportionality calculation.
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