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Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states, for two dimensions, that given a line l and a point P not on l, there is exactly one line through P that does not intersect l, i.e., that is parallel to l. In hyperbolic geometry there are at least two distinct lines through P which do not intersect l, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.
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Computers & Math News
January 26, 2026
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Jan. 17, 2026 Engineers have created a device that generates incredibly tiny, earthquake-like vibrations on a microchip—and it could transform future electronics. Using a new kind of “phonon laser,” the team ...
Jan. 17, 2026 Researchers have turned artificial intelligence into a powerful new lens for understanding why cancer survival rates differ so dramatically around the world. By analyzing cancer data and health system information from 185 countries, the AI model ...
Jan. 16, 2026 Humans pay enormous attention to lips during conversation, and robots have struggled badly to keep up. A new robot developed at Columbia Engineering learned realistic lip movements by watching its own reflection and studying human videos online. ...
Jan. 15, 2026 A new OLED design can stretch dramatically while staying bright, solving a problem that has long limited flexible displays. The breakthrough comes from pairing a highly efficient light-emitting material with tough, transparent MXene-based ...
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Jan. 13, 2026 A generative AI system can now analyze blood cells with greater accuracy and confidence than human experts, detecting subtle signs of diseases like leukemia. It not only spots rare abnormalities but also recognizes its own uncertainty, making it a ...
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Jan. 11, 2026 Scientists have unveiled a new way to capture ultra-sharp optical images without lenses or painstaking alignment. The approach uses multiple sensors to collect raw light patterns independently, then ...
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Jan. 6, 2026 Researchers have created microscopic robots so small they’re barely visible, yet smart enough to sense, decide, and move completely on their own. Powered by light and equipped with tiny computers, ...
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Jan. 3, 2026 Scientists have developed molecular devices that can switch roles, behaving as memory, logic, or learning elements within the same structure. The ...
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