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Seven Skies
Joined October 2018
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The Coding Bat 🐧 retweeted
1 + 3 + 5 + ⋯ + (2n − 1) = n². Each odd integer 2k − 1 appears as a single L of identically colored dots that wraps two adjacent sides of the preceding square.
These layers accumulate until the side length equals n. For n = 7 the finished construction is a 7 × 7 grid that contains exactly 49 dots.
The Coding Bat 🐧 retweeted
Django Rest Framework Release 3.18.0 #python github.com/encode/django-res…
The Coding Bat 🐧 retweeted
It took Russel and Whitehead (1913) almost all of their most famous paper—397 pages—to prove this theorem.
The Coding Bat 🐧 retweeted
When you get a ring notifications in the middle of the night..🐾😅 xitter.cf/Yoda4ever/status/20922…
The Coding Bat 🐧 retweeted
No heat engine can turn all its heat into useful work. Sadi Carnot understood this back in 1824.
He showed that the maximum possible efficiency is set only by the hot and cold reservoir temperatures:
η = 1 − Tc/Th
where Th is the hot reservoir temperature and Tc is the cold one, both measured in kelvin.
The important idea is simple: a heat engine’s ultimate efficiency is limited by its temperature difference, not by how clever its design is.
The Coding Bat 🐧 retweeted
most discoveries begin when someone notices what everyone else learned to ignore ✍️
The Coding Bat 🐧 retweeted
Add squares of side 1, ½, ⅓, ¼ … forever. Their areas sum to a circle constant, not a messy decimal:
1 + 1/4 + 1/9 + 1/16 + ⋯ = π²/6
∑ₙ₌₁^∞ 1/n² = π²/6
The same number is the probability that two random integers are coprime: 6/π².
Geometry of the circle sits inside ordinary counting.
The Coding Bat 🐧 retweeted
Bonnet Theorem ✍️
Every ship that has ever floated every submarine that has ever dived every fish that has ever hovered motionless in water. All of them are balanced by a force whose precise mathematical description took centuries to get exactly right and the Bonnet theorem is the story of where that force comes from and exactly where it acts.
Water pushes on every surface it touches. This is not an observation. You feel it every time you push your hand into a pool.. The precise nature of that pushing is more subtle than it first appears. Water does not push equally in all directions or with strength everywhere. The deeper you go the greater the pressure. Because more water is stacked above and all that waters weight must be supported. At any point underwater the pressure acts equally in all directions. Not downward but sideways, upward, diagonally. A fundamental property of fluids called Pascals principle. This means that every tiny patch of a submerged objects surface feels a pressure force pushing inward. To that surface patch directed toward the interior of the object. And the strength of that push depends precisely on how deep that particular patch is located.
Full explanation given below 👇
The Coding Bat 🐧 retweeted
🐍 Ruff: A Modern Python Linter for Error-Free and Maintainable Code — realpython.com/ruff-python/
#python
The Coding Bat 🐧 retweeted
A path can bend through space yet feel perfectly straight to an ant walking the surface.
That turning inside the tangent plane is geodesic curvature k_g. When k_g = 0 the curve is a geodesic; the straightest possible route on S.
Its space curvature k then satisfies
k² = k_g² + k_n²
Robot motion, mesh processing and relativistic orbits all compute these zero-k_g curves.
Straightness is always measured against the geometry you inhabit.
The Coding Bat 🐧 retweeted
One of the most useful ideas in probability is that overlap matters.
If two events can happen together, simply adding their probabilities counts the overlap twice. That is why
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The same logic leads to conditional probability and, eventually, Bayes’ theorem: knowing how events overlap lets us update probabilities when new evidence appears.
