#YangHuiTriangle
LightNews LightNews
paula-projekt.bsky.social
It is. #Mathematics is magical. Start with the famous #YangHuiTriangle or #PascalTriangle and easily work through a huge part of mathematics, f.e. n-dimensional cubes, curve analysis and calculus of integer polynomials, set theory, and probability theory! See QP for more, please: #infodump alert ; )
The screenshot, which is actually a real screenshot of a computer screen, shows a Queen of the Night flower. A glass plate with a cute formula for Pascal's triangle was placed in front of it. I started with a^n+b^n=c^n with c=a+d and did a cheap trick: I defined b as dm and got a^n+ d^n times the sum of the products of the binomial coefficients and (a/d)^n-k: binomial coefficients (source Wikipedia: https://en.wikipedia.org/wiki/Binomial_coefficient )( n k ) = n × ( n − 1 ) × ⋯ × ( n − k + 1 ) k × ( k − 1 ) × ⋯ × 1 , {\displaystyle {\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},}
August 24, 2025 at 2:30 PM
paula-projekt.bsky.social
I'm so sorry for a mistake!
144²+17²=145² equals 144² +2*144+1*1 with
a²+2ad+d²=(a+d)² and 144 as "a", 1 as "d")
#sharingisthenewlearning #SharingIsCaring #education #mathematics #math #maths #geometry #YangHuiTriangle #PascalTriangle
paula-projekt.bsky.social Paul Arnheim-Projekt @paula-projekt.bsky.social · Jul 31
How To #Polynomial W/O #Epsilon & More
With a tricky formula, the 1st Shalan theorem:
(d(m²-1)/2)²+d²m²=(d(m²+1)/2)²
f.e. d(m²+1)/2=17 d=2,m=4
15²+8²=17²
d,m=1,17
244²+17²=245²
#sharingisthenewlearning #SharingIsCaring #education #mathematics #math #maths #geometry #YangHuiTriangle #PascalTriangle
paula-projekt.bsky.social Paul Arnheim-Projekt @paula-projekt.bsky.social · Jan 15
How To Polynomial And More
Students ask me why the stuff I give them isn't in the textbook even though it's so obvious. My answer: “Not yet. A textbook is in progress.”
You can approach the discussion of curves and the integral calculation of polynomials without epsilon and purely geometrically!
July 31, 2025 at 1:40 PM
paula-projekt.bsky.social
How To #Polynomial W/O #Epsilon & More
With a tricky formula, the 1st Shalan theorem:
(d(m²-1)/2)²+d²m²=(d(m²+1)/2)²
f.e. d(m²+1)/2=17 d=2,m=4
15²+8²=17²
d,m=1,17
244²+17²=245²
#sharingisthenewlearning #SharingIsCaring #education #mathematics #math #maths #geometry #YangHuiTriangle #PascalTriangle
paula-projekt.bsky.social Paul Arnheim-Projekt @paula-projekt.bsky.social · Jan 15
How To Polynomial And More
Students ask me why the stuff I give them isn't in the textbook even though it's so obvious. My answer: “Not yet. A textbook is in progress.”
You can approach the discussion of curves and the integral calculation of polynomials without epsilon and purely geometrically!
Visualization of the famous Pascal or Yang Hui or Tartaglia or ... Triangle with LEGOs, dimensions 0 (point), 1 (edge + edge), 2 (square + two rectangles + square), 3 (cube +3 wall cuboids + 3 edge cuboids + cube) Background: a literally screenshot of a photo that shows epiphyllum oxypetalum. In the foreground there is a formula for the tricky triangle a^n+d^n times the sum of of the binomialcoefficients n!/(n-k)!k! starting with k=1 times (a/d)^(n-k) equals (a+d)^n By outsourcing of d, the difference between c and a (a^n+b^n=c^n, a+d=c), you get the Shalan theorems (see bio or pinned tweet, please). This LEGO cuboid that represents a polynomial of the third degree: f(x)=(x+0)(x+3)(x-1) with x+0 in + x-direction: forward is positive, x+3 in + y-direction: right is positive, 1 in - z-direction: down is negative. =x³+2x²-3x It is a blue cube x³ plus a red cuboid 3x² minus a white cuboid 1x² minus a white cuboid x*3*1. Half of its boundaries, three blue squares x², two red rectangles 3x, two white rectangles -1x, a white rectange 3*-1 equal the first derivative f'(x) of the function f(x) with f'(x)=3x²+(6-2)x-3=3x²+4x-3. It is shockingly simple, isn't it? The graphs were drawn with this ingenious toy/tool: https://www.matheretter.de/rechner/gfplot f(x)=x³+2x²-3x (blue graph) f'(x)=3x²+4x-3 (pink graph) 1st derivative, describes the slope of f(x) f''(x)=6x+4 (turquoise graph) 2nd derivative, describes the slope of f'(x) f'''(x)=6 (green graph) 3rd derivative, describes the slope of f''(x) F(x)=x⁴/4+2x³/3-3x²/2 (purple graph), Antiderivative of f(x), describes the area between the blue graph f(x) and the x-axis. One point describes half of the boundaries of an edge, one edge x describes only the half of half of the boundaries of a square, one x² describes just a third of half the boundaries of a cube, one cube describes just a fourth of half the boundaries of a four-dimensional cube. It really works without epsilon!!! Bonus: You can find the x-value of the vertex or the middle inflection point of any polynomial without a calculator. f(x)=a1x^n+a2x^(n-1)+a3x^(n-2)...a(n+1)x⁰ Vertex, if n is even or middle inflection point, if n is odd: -a2/na1. The x-value of the inflection point of f(x)=1x³+2x²-3x equals -2/3*1=-2/3. This one is lazybones's darling ; )
July 31, 2025 at 10:39 AM