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The first two rowsrow of this table areis straightforward. The homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>0</sup>)}} of the 0-dimensional sphere are trivial for {{math|''i'' > 0}}, because any base point preserving map from an {{mvar|i}}-sphere to a 0-sphere ''is'' a one-point mapping. Similarly, the homotopy groups {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} of the 1-sphere are trivial for {{math|''i'' > 1}}, because the universal [[covering space|universal covering space]], <math>\mathbb{R}</math>, which has the same higher homotopy groups, is contractible.{{cn|date=February 2022}}
Beyond thesethe twofirst rowsrow, the higher homotopy groups ({{math|''i'' > ''n''}}) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
* The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
* Most of the groups are finite. The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow).
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