Here's a nice graph.

"Yes, Nicholas, but what does it mean?"

Let me tell you what it means.

Along the bottom you will see the names, in order, of the US Presidents to date.

Immediately above them is a blue line, which corresponds to the length of each president's term in days. (Counting Cleveland's two terms separately, and assuming Obama serves until 20 January 2017.)

Above the blue line is an orange line, showing the cumulative terms of pairs of presidents who served successively. So the leftmost point corresponds to the almost 12 years, 4,325 days, served by Washington and Adams; the next point is at almost the same level, corresponding to the twelve years exactly, 4,382 days, served by Adams and Jefferson. (Usually there are 4,383 days in twelve years, but 1800 was not a leap year.) And so it continues, dipping to four years for Harrison/Tyler, Taylor/Fillmore and Garfield/Arthur, rising to twenty years for FDR/Truman, and ending with a stable sixteen years for Clinton/Bush and Bush/Obama.

The third line, which is grey, combines each group of three successive presidential terms – from almost 20 years for Washington/Adams/Jefferson to 24 years for Clinton/Bush/Obama, peaking at twenty-eight for FDR/Truman/Eisenhower, dipping to eight three times.

And so on up to the top.

The two presidencies of less than a year – the elder Harrison and Garfield – have the most noticeable effect on the graph, sharp jumps rippling up through the centuries. There are similar but shallower gradients associated with the other short presidencies – Taylor/Fillmore, Harding, Ford, Kennedy.

But if you look closer, you can also see the impact of timing. Both Harrison and Garfield died less than a year into office; both succeeded one-term presidents (Van Buren and Hayes); both of them succeeded eight-year presidents (Jackson and Grant, who had other things in common). So in both cases the gradient shows a sharp decline/increase as those factors hit. By contrast, although Taylor and Fillmore both served less than four years, they were in the middle of a period of one-term presidents and therefore stand out less.

The shallower ripple from FDR's 12 years is also visible – boosted by the fact that his immediate successors also served relatively lengthy terms, but both before (Hoover/Coolidge/Harding) and after (Kennedy/Johnson/Nixon/Ford/Carter) there were periods of shorter service.

I'm sure this can be made more beautiful with someone who has better graphic skills than I do. If you decide to have a go, try this one as well:

This is the inverse of the other graph, in a way. I've taken each quadrennial period, from inauguration to inauguration (thus counting the shorter 1789-93 and 1933-37 periods equally with the rest) and calculated what fraction of a presidential term fitted into each. So, for the 1841-45, 1849-53 and 1881-85 quadrennia, the lowest line, blue again, peaks at 2 presidential terms; FDR's first term, from 4 March 1933 until 20 January 1937 (the date of Inauguration Day was changed by the Twentieth Amendment), was 32.1% of his entire time in office and so represents the lowest dip, fractionally below the 33.0% of the 1937-41 and 1941-45 periods. The next line up shows how many presidents served in each eight-year period, starting at 1 (Washington, 1789-1797) and ending at 1 (Obama, 2009-2017) and dipping to 0.651 (FDR, 1933-41).

You can see the ripple effect of the lengthy presidencies from 1933 to 1961 going right up through the graph. The longer presidential terms both at the start and the end of the overall peiod from 1789 to 2017 also slim down the contours at the edges. Of the shorter presidencies, the Garfield/Arthur period of 1881-1885 stands out a bit more than the earlier ones, partly just because it is closer to the middle I suppose.

Unlike the earlier graph, there are some striking horizontal lines here. The brown line seventh from the bottom, representing 32 years, is almost flat from 1845-1877 to 1881-1913, with eight full presidencies in each of the 32-year periods 1845-77 (Polk-Grant), 1849-81 (Taylor-Hayes), 1853-85 (Pierce-Arthur), 1857-89 (Buchanan-Cleveland 1), 1861-93 (Lincoln-B Harrison), a bit more than eight in 1865-1897 (Lincoln served 3% of his presidency after the 1865 inauguration, then there are eight from Johnson to Cleveland 2), a bit less in 1869-1901 (Grant-Cleveland 2 is seven, and McKinley served 88% of his presidency before the 1901 inauguration), a sliver less in 1873-1905 (half of Grant's term, seven from Hayes to McKinley, 46% of Theodore Roosevelt's term), and eight exactly again in 1877-1909 (Hayes-T Roosevelt) and 1881-1913 (Garfield-Taft). There are others further up (notably near the top, where the 8,4,8,8,8 pattern of the first five presidencies is matched by the most recent five, so Washington-Carter, J Adams-Reagan, Jefferson-GHW Bush, Madison-Clinton, Monroe-GW Bush and JQ Adams-Obama are all periods of 192 years with 39 full presidential terms).

Again, I suspect that someone can present this more beautifully than I have done. If you do, please give me credit!