Linked circles that never meet still fill ordinary space. That is the Hopf fibration: each point of a 2-sphere labels a circle in the 3-sphere, and stereographic projection lays those circles onto nested tori. Any two circles link exactly once. The same linking stabilizes magnetic hopfions, knotted electromagnetic pulses, and light that carries orbital angular momentum. A four-dimensional fact becomes a design rule.

Green circles of alfalfa now dot the golden sands of the Taklimakan Desert in northwest China. Thanks to center-pivot water-saving irrigation, once-barren dunes have been transformed into fertile forage fields, securing a stable feed supply while restoring ecosystems and creating new opportunities for local communities.

The symbolism of “circles” woven into traditional Chinese aesthetics represents the ultimate romance for the Chinese people! Stay tuned on Sep 24th: #AdventuresonMidAutumnFestival2026#

All the red circles on this map have a 500 mile (about 800km) radius. Basic geography knowledge but lots of folks apparently struggle with the logic. On Mercator projections regions closer to the poles appear enlarged. Source:

Number lines pair open circles with strict inequalities (excluding endpoints) and filled circles with inclusive ones, converting exactly to interval notation via parentheses or brackets. x > 5 becomes (5, ∞) x ≥ 5 becomes [5, ∞) 1 < x < 5 becomes (1, 5) 1 ≤ x ≤ 5 becomes [1, 5] x ≤ 1 or x ≥ 5 becomes (−∞, 1] ∪ [5, ∞) In order to activate cooling systems, engineers set temperature safety thresholds in industrial sensors as 1 ≤ T ≤ 5 °C above baseline.

it is funny how life works in circles. When I was studying economics, I was thinking about a doctorate. Then I forgot about it completely. Now, while I'm working on the teaching materials for my new job, this old passion is starting to arise within once again.