J-pEgG🇬🇧ZKEVM🥷03/01/09Protocols are not app's👊

The most beautiful thing about science is that it doesn't claim to know all the answers.🧠

Remember superposition is neither here nor there, until observed✨.👀

[J-pEgG🇬🇧ZKEVM🥷03/01/09Protocols are not app's👊 @CryptoEGG4]

Crypto abstraction!

Thing's need to be explained better!

Instead of being manipulated for one's own financial gain!

So here's

A few handy tools I've come across to lower the barrier's to entry for newcomers, for a overall better user experience!

🧵👇
https://t.co/74HCCicyZ3

[growthepie 🥧📏 @growthepie_eth]

📣 Live now: growthepie Blockspace Explorer.

🧱 Get an overview of the blockspace usage of different Layer-2s
🗂️ Compare the usage of a Layer-2 using different categories
📑 See the most active contracts for each chain and category

How? See for yourself 👇

[Stellarix @Stellarixorine]

Mathematics is a powerful weapon that change the world.

[Cliff Pickover @pickover]

Mathematics, physics.

Creating "teardrops" using mechanical linkages (an assembly of systems connected to manage forces and movement; the link is considered to be rigid).

By上木 敬士郎 / Keishiro Ueki, @KeishiroUeki, Used with permission.

[Mathelirium @mathelirium]

One Particle, Two Slits, One Wavefunction

A double-slit pattern comes out of the time-dependent Schrödinger equation.

In this episode we evolve a single 2D wavepacket with a split-step Fourier solver, send it into a hard barrier with two narrow openings, and render the actual probability density |ψ(x,y,t)|² as a moving 3D surface, with the surface colours tied to the phase arg ψ so the interference pattern is coming straight from the complex wavefunction.

#QuantumMechanics #DoubleSlit #SchrodingerEquation #Wavefunction #PhysicsAnimation #ScientificVisualization

[Mathematica @mathemetica]

Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.

— Archimedes

[Mathelirium @mathelirium]

Here, we took the same roots and poles from our 2D Blaschke-Quotient artwork (see below), wrapped the complex plane onto the Riemann sphere, and let the field bend the surface into a rotating 3D mathematical sculpture.

[Cliff Pickover @pickover]

Math, Universe, Beauty.

Complex Polynomial. Particles ride level curves. Roots appear as raised knobs. Imagine showing this visualization to the greatest mathematicians of the 1800s.

By @mathelirium, https://t.co/jvwFjCtWbI, Used by permission.

[Cliff Pickover @pickover]

FREE Math Book. 630 pages. A gateway into the architecture of infinity, proof, and the future of human thought.

"An Infinite Descent into Pure Mathematics" by Newstead. Topics: prime numbers, logic, sets, functions, induction, relations and infinity, mathematical proofs, number theory, combinatorics, real analysis, infinite cardinality, and probability theory. Undergraduate-level textbook. "I am grateful to the 1000+ students who have already used this book to learn mathematics. The goal of this book is to help the reader make the transition from being a consumer of mathematics to a producer of it."

Link: https://t.co/Bu83aMKVJT

[Mathelirium @mathelirium]

The Torus Becomes the Canvas

A Jacobi Theta Function lives naturally on a complex torus.

Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry.

What you are see is a periodic quantum-like field painted onto the space where it actually belongs.

#JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

[Cliff Pickover @pickover]

FREE Math Book.

"Applied Combinatorics," by Keller & Trotter. "We show students the beauty of combinatorics and how combinatorial problems naturally arise in many settings.... Combinatorial mathematics is one of the most fascinating and captivating subjects on the planet. Combinatorics is very concrete and has a wide range of applications, but it also has an intellectually appealing theoretical side. Our goal is to give you a taste of both."
Contents:
An Introduction to Combinatorics
Strings, Sets, and Binomial Coefficients
Induction
Combinatorial Basics
Graph Theory
Partially Ordered Sets
Inclusion-Exclusion
Generating Functions
Recurrence Equations
Probability
Applying Probability to Combinatorics
Graph Algorithms
Network Flows
Combinatorial Applications of Network Flows
Pólya’s Enumeration Theorem
The Many Faces of Combinatorics

Link: https://t.co/tvT0ADT5yo

[Mathelirium @mathelirium]

The Wavefunction Learns the Shape of a Barrier

A Quantum wavepacket moves across a two-dimensional plane and meets a smooth potential island.

The surface is the computed probability density |ψ(x,y,t)|², while the colour comes from the phase arg ψ. The ripples, transmitted lobes, reflected interference, and tunneling leakage all come directly from the Schrödinger equation.

#QuantumScattering #PotentialBarrier #ProbabilityDensity #WavefunctionPhase #QuantumTunneling #SchrodingerEquation

[Mathelirium @mathelirium]

The Lemniscate Forge

A moving Complex Polynomial draws invisible Lemniscates across the plane, and the particles ride those level curves like bright metal filings caught in an algebraic current. The roots appear as raised 2.5D knobs, while the field folds around them in velvet bands of gold, copper, ivory, and deep mineral shadow.

[Ethereum on ARM (and RISC-V) 🦇🔊🐼👉👈🐼 @EthereumOnARM]

Totally impressed with the @ethrex_client Execution Layer client:

✅ 500 GB database
✅ Minimal CPU load
✅ Rock solid stability

Running it in our @Obol_Collective cluster for a month now.

Outstanding work. 🙌 #Ethereum

[Mathematica @mathemetica]

Fractals: The Hidden Mathematics of Nature’s Infinite Beauty 🧵

Have you ever noticed how a tree branch looks like a smaller version of the whole tree? Or how a coastline looks equally jagged whether you zoom in or zoom out?

This is the world of Fractals; shapes that repeat their patterns at every scale. (1/5)

[Cliff Pickover @pickover]

Behold the bacterial flagellar motor assembly. It's amazing to realize that this spontaneously appeared, after billions of years, from the quark-gluon plasma of the Big Bang. The universe somehow taught matter to build nanoscale engines.

Figure source: https://t.co/aQJucaUc87

[Mathelirium @mathelirium]

When Software Touches The World, Latency Becomes Physics

In the age of AI, it is tempting to think that programming is becoming less important because large language models can write code, debug code, and even help design algorithms.

But the moment software touches hardware, the problem changes.

#AIHardware
#Robotics
#ControlSystems
#EmbeddedAI
#PhysicalComputing
#EngineeringReality

[Cliff Pickover @pickover]

Mathematics. It's alive!

In cellular automata, such as Conway's Game of Life, a Breeder is a pattern that generates copies of a secondary pattern, each of which then generates copies of a tertiary pattern. https://t.co/sSkFlwn6k9

[Mathematica @mathemetica]

PHYSICS

By W. Edward Gettys, Frederick J. Keller, Malcolm J. Skove

“Designed for the calculus-based introductory physics course for engineers and science majors, this edition continues its student-oriented approach with a clear, specific-to-general development of concepts and functional use of full color throughout.”

— W. Edward Gettys, Frederick J. Keller & Malcolm J. Skove

[Mathematica @mathemetica]

Quantum operators are the mathematical tools that turn wave functions into predictions of particle behavior at atomic scales.

The essentials:
> position f(x) (any function like potential V(x)),
> momentum pₓ = ħ/i ∂/∂x (y and z identical in form)
> time-independent Hamiltonian E = p_op²/2m + V(x), > time-dependent E = iħ ∂/∂t, kinetic energy KE = -ħ²/2m ∂²/∂x², and z-angular momentum L_z = -iħ ∂/∂φ.

They drive semiconductor chips in electronics, lasers, MRI scanners, and quantum computers.

[Mathematica @mathemetica]

The Koch curve is constructed iteratively from a straight line segment.

Order 0 (Initiator): straight line, length = 1
Order 1 (Generator): length = 4/3
Order 2: length = 16/9
Order 3: length = 64/27

With each order, every straight segment is replaced by four segments each one-third as long, multiplying the total length by 4/3. As n → ∞, length → ∞.

[Mathematica @mathemetica]

Calculus notation decoded: integrals accumulate quantities while derivatives capture instantaneous rates of change.

The integral ∫_a^b f(x) dx sums the integrand f(x) from lower limit a to upper limit b, with x the dummy variable of integration. Evaluating it yields the net value over that interval.
The derivative df/dt equals the limit as h→0 of [f(t + h) − f(t)] / h, defining the rate of change of the quantity with respect to time as the interval h tends to zero.

These tools drive real-life work like modeling rocket trajectories in aerospace and predicting economic trends.