Vatsal Bakshi

I work on the Crypto Services team at Apple, a smaller and specialized group embedded within the Information Systems and Technology (IS&T) organization. Before this I was working on engineering challenges associated with infrastructure in data centers supporting various compliance and security teams.

I am a hands-on engineering leader who enjoys solving software engineering problems at any scale.

I received my Masters in Information Security from Johns Hopkins University and my B.E. in Computer Science from Birla Institute of Technology & Science, Pilani.

Away from the keyboard, I spend time outside golfing or hiking with my wife. I'm also an avid astrophotographer — some of my work can be viewed on Astrobin.

A mathematician named Klein
Thought the Möbius band was divine.
     Said he: "If you glue
     The edges of two,
You'll get a weird bottle like mine."

— Leo Moser

Klein bottle ↗ · 3Blue1Brown ↗ ·

Parametrization

Figure-8 immersion (Apéry, 1984)

r(u) = 2\!\left(1 - \dfrac{\cos u}{2}\right), \quad u \in [0,2\pi],\; v \in [0,2\pi]

u < \pi:\quad \begin{cases} x = 3\cos u\,(1+\sin u) + r(u)\cos u\cos v \\ y = 8\sin u + r(u)\sin u\cos v \\ z = r(u)\sin v \end{cases}

u \ge \pi:\quad \begin{cases} x = 3\cos u\,(1+\sin u) + r(u)\cos(v+\pi) \\ y = 8\sin u \\ z = r(u)\sin v \end{cases}

Three.js

Glass shell + concentric liquid geometry, both from the same parametric function.

function makeKlein(tubeScale, v, u, target) {
  u *= Math.PI; v *= 2 * Math.PI; u *= 2;
  const r = 2 * (1 - Math.cos(u) / 2) * tubeScale;
  if (u < Math.PI) {
    target.set(
      3*Math.cos(u)*(1+Math.sin(u)) + r*Math.cos(u)*Math.cos(v),
      8*Math.sin(u)                 + r*Math.sin(u)*Math.cos(v),
      r * Math.sin(v)
    );
  } else {
    target.set(
      3*Math.cos(u)*(1+Math.sin(u)) + r*Math.cos(v + Math.PI),
      8*Math.sin(u),
      r * Math.sin(v)
    );
  }
}

// Glass shell — tubeScale 1.0
new MeshPhysicalMaterial({
  color:           0xe8f0ff,
  transmission:    0.88,
  thickness:       2.5,
  ior:             1.52,
  reflectivity:    0.65,
  roughness:       0.0,
  envMapIntensity: 2.5,
  side:            DoubleSide,
});

// Liquid fill — same surface at 72% tube radius
new MeshStandardMaterial({
  color:     0x07070f,
  metalness: 0.2,
  roughness: 0.6,
  side:      FrontSide,
});

"evi opera dedicator meansura vasculie,
pondere non magni, quod interim helluo nullus ebibat"

(I give the measurements of a drinking glass (or vase), that has a small weight, but that even the hardest drinker could not empty.)

— de Sluse in a letter to Huygens, translation Jan A. van Maanen

Gabriel's horn ↗ · Numberphile ↗ ·

Parametrization

Surface of revolution of y = 1/x about the x-axis (Torricelli, 1643)

y = \frac{1}{x} \text{ revolved about the } x\text{-axis},\quad x \in [1, \infty)

u \in [1,\infty),\quad v \in [0, 2\pi] \qquad x(u,v) = u,\quad y(u,v) = \frac{\cos v}{u},\quad z(u,v) = \frac{\sin v}{u}

\text{Volume} = \pi \text{ (finite)}, \qquad \text{Surface area} = \infty

Three.js

Single parametric mesh; logarithmic u-distribution concentrates detail at the opening.

function makeHorn(u, v, target) {
  // t ∈ [1, 40] — logarithmic distribution, needle-thin at tip
  const t = Math.pow(40, u);
  const r = 1 / t;
  const angle = v * 2 * Math.PI;
  target.set(t, r * Math.cos(angle), r * Math.sin(angle));
}

// Alpha gradient: opaque at opening (u=0), transparent at tip (u=1)
const ac = document.createElement('canvas');
ac.width = 256; ac.height = 4;
const gCtx = ac.getContext('2d');
const grad = gCtx.createLinearGradient(0, 0, 256, 0);
grad.addColorStop(0.0, 'white');
grad.addColorStop(0.5, 'white');
grad.addColorStop(1.0, 'black');
gCtx.fillStyle = grad;
gCtx.fillRect(0, 0, 256, 4);

new MeshStandardMaterial({
  color:       0xb07d3a,
  metalness:   0.85,
  roughness:   0.18,
  alphaMap:    alphaTex,
  transparent: true,
  side:        DoubleSide,
});