A mathematical construction in which one strip completes exactly... ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.show more

Euler's formula states that e^(iθ) = cos θ +... i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.show more

A radius turns at a steady rate inside a... regular hexagon, a decagon and a circle, and the height y = r(θ) sin θ of its tip is traced against time. For an n-gon of circumradius R with a vertex at θ = 0, r(θ) = R cos(π/n) / cos((θ mod 2π/n) − π/n). Each edge is one piece of the wave; the flat top edge is a plateau at R cos(π/n), 0.866R for n = 6, 0.951R for n = 10. As n grows the plateaus narrow and rise to R, and the wave tends to R sin θ.show more

A closed outline shaped like the Greek letter pi... can be rebuilt as the path of a point carried by a chain of rotating circles, each circle matching one frequency component through its radius and phase, so that successive circles refine the match to the original curve.show more

A bar magnet spinning at angular speed Ω is... a magnetic dipole m turning in its own plane, and it radiates at f = Ω/2π. In that plane E points straight out of it, E ∝ m̈ × r̂ at the retarded time t − r/c, so each far crest lies on θ = φ(t − r/c) + π/2: two spiral arms of pitch λ = 2πc/Ω. The whine is 1024f, ten octaves up. In empty space J = 0, so in ∇ × B = μ₀ε₀ ∂E/∂t + μ₀J only the changing E keeps B alive. The radiated power grows as Ω⁴.show more

A rose curve is r = sin(kθ). Here k... = b/a, with a = 1, 3, 7 down the rows and b = 2, 4, 8 across. With k = p/q in lowest terms and pq even, the rose has 2p petals and closes after θ = 2qπ. Each a is odd and each b a power of 2, so p = b and q = a: each column has 2b petals, 4, 8 and 16, and a only sets how far they wind: 2π, 6π, 14π. Swept over θ ∈ [0, 2aπ] in equal times, kθ = 2πb·t/T, so a column's three pens reach their petal tips together.show more

3D rotation matrices about coordinate axes. Rₓ(θ) fixes x... and rotates yz-plane [[1 0 0] [0 cosθ −sinθ] [0 sinθ cosθ]]. Rᵧ(θ) fixes y and rotates xz-plane [[cosθ 0 sinθ] [0 1 0] [−sinθ 0 cosθ]]. Rz(θ) fixes z and rotates xy-plane [[cosθ −sinθ 0] [sinθ cosθ 0] [0 0 1]].show more

A turnstile antenna is two crossed dipoles driven 90°... apart, one dipole p turning at ω about its mast. Its far field E ∝ r̂ × (r̂ × p̈(t − r/c)) has phase ωt − kr − φ at every latitude, so each wavefront is the spiral r = (ωt − φ − 2πn)/k. Along the mast E ∝ (1/r)[cos(kr − ωt) ŷ + sin(kr − ωt) x̂] keeps its length and turns once per period, and B = (1/c) r̂ × E turns with it. In the plane of the cross the same wave is linearly polarized.show more

HOW MANY GIRL HUMANOIDS CAN YOU KEEP AT HOME... IS THERE A LIMIT FROM THE GOVERNMENT YOU WILL BE SURPRISED WHO ACTUALLY SETS IT ok wait everyone imagines a law that says "max 2 per person" like a pet limit or a gun license ngl that is the wrong test in most places she's just property you can own one or ten the state doesn't count her but four other things do ▹ limit 1: the battery every one of them runs on a big lithium pack ten of them is ten packs charging in one apartment that's a fire load your building was never planned for insurers, landlords and house rules already fight e-bike batteries in hallways and basements a room full of charging humanoids is that, times ten ▹ limit 2: the floor and the space a walking frame like the unitree g1 is around 35 kg companion bodies with silicone shells are often heavier ten of them is a third of a ton on one floor plus charging spots, spare skin, a repair corner and a box the size of a person for each one ▹ limit 3: the data each one has cameras and microphones inside your home that's mostly your business but the moment they record guests, neighbours or the street you're not just an owner anymore in europe that's where gdpr starts asking questions ▹ limit 4: the business line own ten for yourself: a very expensive hobby rent them out, film content, charge for time with them: now you're a business licenses, taxes, insurance, and in some cases laws you really don't want to be on the wrong side of ▹ the surprise the government doesn't limit how many you own your fuse box, your landlord, your insurer and your wallet do at ¥119.8k+ each, the wallet usually wins first so the real number is as many as your apartment can charge safely at night which for most people is exactly one how many would you keep, honestly tell me in the repliesshow more

YEMEN UAP Three pieces of “debris” (identical IR signatures)... appear to emerge from the object *in succession.* That is inconsistent with a kinetic strike ripping debris off of an object. It’s almost like they were inside the UAP and emerged one-by-one.show more

Another angle of the near train collision of a... school bus in Sumter County, Florida has been posted Earlier footage from inside the school bus showed the kids yelling as they almost got hit Sumter County Sheriff's Office said Yvonne Hampton is accused of 29 counts of child neglect and one count of endangering an adultshow more

#BREAKING Two vehicles have been swept away by fast-moving... floodwaters in Oxford. The rear tail lights of one is being pummeled by the rapids. EMS says 2 people are trapped at the Brookside Inn and another person is trapped inside a home. Eversource is also here WTNH News 8show more

Three artificially intelligent (AI) generated videos showing highly objectionable... visuals of Makkah and Madina, one of the holiest sites of Muslims, surfaced on social media platforms, mainly Instagram, on Friday. Posted by Rohit Kumar, a resident of Akhtiyarpur Manjhauli village in Patna, on his Instagram profile (click here), the video surfaced on Friday, July 17. In one video, Kumar appears in a saffron scarf and enters the Kaaba with a pig, which defecates on the floor. "Duniya mein pehla aadmi hoon, jo Makkah-Madinah mein suar (pig) laaya hoon. (I am the first person who has brought a pig inside Makkah-Madinah)." Another video depicts him entering Makkah-Madinah and showing a shivaling. And the third video shows ancient Hindu statues right inside the holy premises. Rohit Kumar has an Instagram following of 25.8K. He also has another ID (click here) in which he has just seven posts and followers.show more

When a drone applies a force at an angle... to a cart resting on a level surface the force produces a horizontal pull that moves the cart forward and a vertical lift that lightens the load on the wheels, illustrating how vectors separate into independent effects along two perpendicular axes.show more

We live in an illusion, a simulation created by... nonphysical electromagnetic waves collapsing into material reality, where your soul is trapped inside an energetic container that we call matter. There, it learns the lessons of Good and Evil necessary for its evolution. That is why the body is made of earth/dirt/matter, while the soul is made of fire, which is plasma. In Hinduism and Buddhism, this is represented by the Bhavachakra, or Wheel of Life, which depicts samsara, the cycle of death and rebirth in which the soul is “trapped” until it escapes the wheel after acquiring the knowledge of Good and Evil and attaining wisdom. The three poisons at the center of samsara are actually the northern constellations Draco, Cygnus, and Ursa Major, constantly chasing one another in a circle, symbolizing both the passage of time and the endless, repeating cycles. The twelve outer sections represent the twelve zodiac constellations. So yes, we live in an illusion, a simulation, but not an illusion created by AI, as in The Matrix. It is an illusion created by God for the purpose of the soul’s evolution.show more

This is extra innings in an elimination game in... the B1G Tournament. Iowa trails Illinois 4-2 with the bases loaded, no outs. This slide by Kyle Huckstorf is ruled “interference”. Results in a DP & negates a run. One of the worst calls you’ll ever see.show more

One of the most haunting moments of my life... was visiting Auschwitz and standing inside the remaining gas chambers, where hundreds of thousands of Jews and others were murdered in a systematic way over years. What stays with you is not noise. It’s the silence. The emptiness of a place built for industrial killing. You stand there knowing that families were forced inside together, knowing exactly what was about to happen, watching children, parents, siblings, and friends disappear one by one. The walls are quiet now, but your mind fills in what history erased. This is the largest crime scene in human history.show more

Just look at this Axe which the Co-Pilot used... to attack the Captain in the Fly Dubai incident. It’s a heavy-duty emergency axe with a sharp cutting edge, built to break through tough aircraft materials when normal access is blocked. It looks and is like a serious piece of equipment. In the wrong hands, a tool with this kind of weight and cutting edge could cause devastating injuries. And for your information, this tool is a must to have inside an aircraft. Aviation regulations require aircraft with more than 19 passenger seats to carry emergency equipment such as a crash axe. An emergency tool designed to help save lives suddenly became part of an attack inside Fly Dubai cockpit! The UAE Attorney-General confirmed that the co-pilot attacked Captain Smit Machchhar with the aircraft’s crash axe inside the cockpit before attempting to take control of the aircraft. The axe is for an A320 corkpit, not the exact one which was used.show more