Reference Terms
from Wikipedia, the free encyclopedia
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.
Related Stories
1
2
Computers & Math News
August 29, 2026
Aug. 27, 2026 Researchers used AI to search through more than 100 million possible settings for 3D-printing a high-performance NASA alloy. After only 40 experiments, the system identified six successful ...
Aug. 27, 2026 Scientists have found a surprising way to potentially switch on an unusual form of magnetism in ruthenium dioxide, a material that normally appears nonmagnetic. When the material was made into an ...
Aug. 26, 2026 NASA has successfully tested a system that allows satellites to navigate without GPS by using other spacecraft and debris as landmarks in orbit. In just three days, FALCON also improved the known orbits of more than 200 space objects completely ...
Aug. 24, 2026 Researchers have created a tiny chip that produces a stable “rainbow” of light capable of generating multiple high-frequency signals at once, potentially boosting the speed and capacity of future 6G networks. Its extreme precision could also ...
Aug. 24, 2026 More screen time during infancy and around the start of school was linked to poorer academic performance and weaker working memory years later. The findings suggest that early childhood may contain particularly sensitive windows when screen habits ...
Aug. 23, 2026 Researchers have used AI to uncover the DNA signature of a key genetic “switch” involved in turning genes on. After analyzing about 500,000 DNA sequences, the model identified the initiator in roughly 60% of human genes. The breakthrough could ...
Aug. 22, 2026 Freezing the liquid core of an optical fiber produced an extreme environment where light and sound interact more than 1,000 times more strongly than in ordinary fibers. Researchers used the effect to create optoacoustic memory, potentially paving ...
Aug. 21, 2026 Scientists watched a light-triggered hidden state form inside a material in only 30 femtoseconds, revealing a step that had never been seen before. The material first entered a fleeting electronic state in which its bonds reorganized in a repeating ...
Aug. 18, 2026 Researchers have built microscopic, light-driven robots that can rapidly navigate through liquid, collect bacteria, and deposit them in chosen locations. These tiny “cleaners” could open new possibilities for manipulating cells and microbes with ...
Aug. 17, 2026 Caltech scientists have created ultra-low-loss optical pathways on silicon chips that approach the efficiency of fiber optics and dramatically outperform existing technology at visible wavelengths. The breakthrough could unlock more powerful lasers, ...
Aug. 17, 2026 Researchers combined synthetic DNA with a semiconductor to create an ultra-low-power memory device capable of storing and processing information in the same place. The bio-hybrid technology could eventually help make AI systems and next-generation ...
Aug. 17, 2026 An eight-year Finnish study found that children who spent more time on screens tended to show better cognitive processing as teenagers, challenging common assumptions about screen use. Researchers ...
Latest Headlines
updated 12:56 pm ET
Aug. 16, 2026 Scientists have discovered that tiny, sharply curved wrinkles in graphene can dramatically alter its electrical behavior, creating surprisingly strong charge separation. The finding suggests future ...
Aug. 14, 2026 A tiny superconducting engine has successfully converted heat near absolute zero into useful work, demonstrating the first cyclic quantum heat engine ...
Aug. 12, 2026 Ordinary WiFi networks could quietly become powerful surveillance tools, allowing people to be identified without cameras, special sensors, or even carrying a connected device. Researchers showed ...
Aug. 9, 2026 Atomically thin semiconductors could enable dramatically smaller and more efficient chips, but a stubborn problem at the boundary between materials has limited their performance. Researchers have now ...
Aug. 8, 2026 A new nanostructured carbon design lets fuel-cell catalysts use tiny amounts of platinum while remaining remarkably stable and efficient. The breakthrough could help hydrogen fuel cells become a more ...
Aug. 7, 2026 Scientists have generated quantum entanglement directly from sunlight, potentially offering a lower-energy alternative to the lasers normally used in quantum technology. Their outdoor experiment ...
Aug. 5, 2026 A mathematician working at Anthropic says he used the AI model Claude Fable 5 to uncover a remarkably simple counterexample to the Jacobian ...
Aug. 3, 2026 Researchers have found a way to build much larger “twisted” oxide materials while precisely controlling how their atomic layers line up. Because these materials can be made over large areas and ...
Aug. 2, 2026 Mathematicians have shown that no electoral system can perfectly balance local representation, proportional national results, and a fixed-size parliament once enough parties compete. A newly proposed ...
July 31, 2026 A person’s sleeping brain may reveal warning signs of dementia long before memory problems begin. Researchers used machine learning to analyze EEG recordings from about 7,000 adults and found that ...