Bhaki's Engineering Journal

A personal study room

The Euler Room

Following the simple connections Leonhard Euler found inside difficult problems—and the engineering world built on top of them.

A place for the ideas, papers and theorems I want to understand slowly, explain simply and return to often.

A place I genuinely love

Project Euler

I genuinely love Project Euler. It makes mathematics feel active: begin with a small question, write a program, notice a pattern, and finish with an idea that stays with you.

Explore the problems

His mathematics is still running

How Euler quietly shaped the modern world

Not a catalogue of famous formulas. This is a journey from the questions Euler asked to the machines, networks and signals surrounding us today.

01

Euler's idea

Graph structure

Keep the connections. Ignore the distracting shape.

The world built on it

Networks become navigable

Internet routing · dependency graphs · logistics · circuit designRead this topic
02

Euler's idea

Complex exponentials

Connect rotation, oscillation and exponential change.

The world built on it

Waves become manageable

Power grids · radio · audio · signal processingRead this topic
03

Euler's idea

Differential equations

Describe how a system changes from one instant to the next.

The world built on it

Motion becomes predictable

Flight control · robotics · orbital mechanics · simulationRead this topic
04

Euler's idea

Fluid equations

Turn the motion of liquids and gases into mathematics.

The world built on it

Flow becomes computable

Aircraft · turbines · weather forecasts · ocean modelsRead this topic

Four ideas, explained slowly

Start with the question—not the formula

Each idea follows the same path: begin with a real problem, simplify it, understand Euler's move, and then connect it to engineering.

01 · Graph theory

Can I cross every bridge exactly once?

Euler's breakthrough was not finding a clever walking route. It was proving that no such route could exist.

Turning the bridge problem into a graphFour land regions become four nodes. Seven bridges become seven edges. All four nodes have odd degree, so the walk is impossible.1 · KEEP ONLY CONNECTIONSLAND + BRIDGES2 · COUNT EACH NODE35334 ODD NODES → IMPOSSIBLE
  1. Remove the geography.Land regions become nodes. Bridges become edges.
  2. Count each node's degree.The degree is simply the number of connected bridges.
  3. Apply the odd-degree rule.A walk using every edge once can have at most two odd-degree nodes. Königsberg had four.
Modern application · Movement cycles

A Rubik's Cube, flattened into moving cycles

The animation turns the cube into a flat map. Instead of hiding stickers on the back and bottom, it places every sticker position on a visible circular track.

  1. Each colored dot is a sticker position.The flat picture lets us see the cube's six faces at the same time.
  2. Each circle represents a movable layer.The cube has three layers along each of three axes, giving nine layer cycles in this view.
  3. A turn moves several dots together.Watch one cube layer and its matching circular track rotate at the same moment.
One surprising result

However scrambled it looks, every position can be solved in at most 20 moves when a quarter-turn or half-turn of any face counts as one move. The graph-theory version of this statement is: the cube graph's farthest node is 20 edges from solved.

Further reading: how the circular movement map is constructed, the number of cube positions, the proof that 20 moves are enough, and how pattern databases guide a solver.

02 · Complex numbers

Why can an imaginary number describe a real wave?

Euler's formula turns rotation into arithmetic. A rotating point carries cosine on one axis and sine on the other.

Euler's formula connects rotation to wavesA rotating point carries cosine and sine. At pi radians it reaches minus one, producing Euler's identity.θcos θsin θ−1ROTATION WRITTEN AS A NUMBEReⁱθ = cos θ + i sin θSET θ = πeⁱπ + 1 = 0ROTATION · WAVES · EXPONENTIAL CHANGE
  1. Begin with a point on a circle.Its horizontal position is cosine; its vertical position is sine.
  2. Store both values in one complex number.Multiplication can now rotate the point instead of handling two equations.
  3. Rotate by π radians.The point reaches −1, so adding 1 produces Euler's identity.
03 · Differential equations

If I know how something changes, can I predict what happens next?

Euler's method takes the rate of change at the current state and uses it to make one small step into the future.

Predicting change one small step at a timeStart from the current state, calculate its rate, take a small step, and repeat to approximate the future.PREDICT THE FUTURE IN SMALL STEPSSTATE NOWRATESMALL STEPNEXT STATEposition, speedhow it changesrate × Δtnew estimateSTATE NEXT = STATE NOW + RATE × SMALL TIME STEP
  1. Describe the current state.For motion, that might be position and velocity.
  2. Calculate the present rate.The differential equation tells us how the state is changing now.
  3. Take a small step and repeat.Smaller steps usually improve the approximation but require more work.
04 · Fluid mechanics

How can equations describe air and water that never stop moving?

Instead of following every molecule, Euler described velocity and pressure as fields: values defined at every point in space.

Velocity and pressure describe fluid motionStreamlines bend around an obstacle. Euler's equations relate fluid velocity, pressure and external forces at every point.A FIELD: A VALUE AT EVERY POINTvelocity + pressurevelocity + pressureOBSTACLEPRESSURE GRADIENT + FORCES → CHANGING VELOCITY
  1. Describe the field.Every location has a velocity, pressure and density.
  2. Apply conservation laws.Mass and momentum cannot disappear as the fluid moves.
  3. Compute how the field evolves.Pressure gradients and external forces change the velocity.

The reading shelf

Papers, theorems and my notes

Each entry begins with the original question, builds the idea visually, and ends with what an engineer should remember.

1736 · Graph theory

The bridges of Königsberg

From a city walk to the language of networks.

Modern application · Movement cycles

A Rubik's Cube, flattened into moving cycles

Watch a layer turn on the cube and on a flat map at the same time.

1748 · Analysis

Euler's identity, one connection at a time

Why exponentials, rotation and waves meet in one line.

1757 · Fluids

Making invisible flow understandable

From ideal fluids to aircraft and weather prediction.