the first instance of the integral symbol, excerpt from Leibniz's manuscript
∫∫∫ INTEGRAL DAY ∫∫∫
The integral symbol is 350 years old! On October 29th, 1675 Leibniz has written down the first ever integral sign (see the image).
Let's celebrate integral day by sharing your favourite integral or integral related fact / content.
“springs, strings, airplane wings
steel beams, light beams, and water streams
building sways, ocean waves, and sound waves...”
— Peter Lax (1926-2025)
Planets ~ Springs | The Newton-Hooke Duality and Beyond tamasgorbe.wordpress.com/2025/05/09/p...
Me (15 years ago):
Definitions < Theorems
Me (now):
Definitions > Theorems
Happy Pi Day everyone!
I measured π using a planimeter to celebrate #PiDay
youtu.be/1M0d2Hw9r10
Science is prose, Mathematics is poetry.
The inverse trig functions arcsin and arccos as actual arc lengths
My favourite matrix identity
the determinant of the exponential equals the exponential of the trace
Fibonacci Numbers via Matrix Multiplication
Gabriel's Horn is a solid you get by rotating the hyperbola y=1/x (with x>1) about the x-axis.
Having finite volume (π) and infinite(!) surface area, it leads to the apparent paradox:
"You can fill it with paint, but you cannot coat it."
Pascal's Determinant
Any square matrix cut from the top corner of Pascal's Triangle has determinant 1.
Consider the function
f(x) = 1 / ( ⌊x⌋ + 1 − {x} )
and the sequence of numbers
0, f(0), −f(0), f(f(0)), −f(f(0)), f(f(f(0))), −f(f(f(0))), ...
Congratulations, you've just listed every rational number exactly once!
Trigonometric Addition Formulas
− a visual proof −
sin(α+β) = sin(α)cos(β) + cos(α)sin(β)
cos(α+β) = cos(α)cos(β) – sin(α)sin(β)
Everyone knows that all circles are similar. But did you know that all parabolas are similar?
The ratio of the red arc and the blue focal segment is
√2 + ln(1+√2) = 2.29558...
for every parabola.
This is the universal parabolic constant, the “π of parabolas”.
