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Blaise Pascal was a mathematician and natural philosopher whose scientific work covered projective geometry, probability theory and hydrostatics. His correspondence with Pierre de Fermat in 1654 helped establish methods for calculating chances and dividing the stakes of an interrupted game. He also studied atmospheric pressure and developed a mechanical calculating machine.^[ref] The Pascal triangle connects these interests through binomial coefficients: its addition rule counts choices according to whether a specified object is selected. The same coefficients describe the distribution of the number of successes in repeated independent trials with a fixed success probability.

Geometry studies shapes, positions and spatial relationships through the structures and transformations under consideration. Euclidean geometry measures lengths and angles; projective geometry emphasizes incidence; differential geometry studies smooth spaces using tangent spaces and curvature. A circle can be stretched into an ellipse by an affine map, but generally not by a Euclidean isometry. Their curves are also homeomorphic, illustrating how topology preserves a different level of structure.

In a graded ring $R={⨁}_{d\ge 0}{R}_d$, a homogeneous element of degree $d$ is an element of the component $R_d$. Every ring element is a finite sum of homogeneous components. A nonzero homogeneous element has a unique degree because the sum is direct; zero belongs to every component and usually has no unique assigned degree.

In $k[x,y]$ with total-degree grading, $x^2+3xy-y^2$ is homogeneous of degree two, although it has three terms. The element $x^2+y$ is not homogeneous: its nonzero components lie in degrees two and one. The grading rule gives $R_d{R}_e\subseteq R_{d+e}$, so a nonzero product of homogeneous elements has degree $d+e$.

Grading depends on the declared convention. Assigning weight two to $x$ and weight one to $y$ makes $x+y^2$ homogeneous of weighted degree two. Homogeneity organizes computations in graded modules and the equations used in projective geometry.