Try Exploring Vector in Hexagon and Heptagon with Excel

Try exploring Vector in Hexagon and Heptagon with Excel.
Excel Vector Trigonometry is launched today to show how to use Excel to make calculation for vectors with trigonometry.
- November 26, 2006 -

06/12/14
"<-0.5: The minimum inner product with T, O, B, all on the circle with radius 1>" is shown on the spreadsheet at "Vector in Hexagon", Excel Vector Trigonometry.

06/12/21
"<4: The maximum inner product with T, O, B, all on the circle with radius 1>"

Vector in Octagon

sqrt(2) - sqrt(8 * (sin(22.5 degrees)^2) * (1 - cos(135 degrees)))
= 0

It shows that the length of the shortest diagonal line of an octagon inscribed in a circle with radius of 1 is square root of 2. How cosine formula and inner product work with vectors in octagon is demonstrated in my spreadsheet.

'Vector in Pentagon' was added to my spreadsheet to calculate magnitude of vector on the diagonal line of a pentagon which is a side of pentagon times phi. (06/11/25)

Phi Figures

(5 / (1 / the golden ratio)) - (5 / the golden ratio) = 5

(10 / (1 / the golden ratio)) - (10 / the golden ratio) = 10

"Phi Figures" - a set of figures calculated with phi
and trigonometric function - is added on this spreadsheet.
- November 21, 2006 -

Phi^10-by-Phi^10 Rectangle and Progressive Difference

(phi^20) - (1 + phi + (phi^3) + (phi^5) + (phi^7) + (phi^9) + (phi^11) + (phi^13) + (phi^15) + (phi^17) + (phi^19)) = 0 *1 *2 shows how I am going to theorize my "Phi-by-Phi (Golden) Rectangle" with Excel Trigonometry.

Excel, Google and websites like Virtual Library of Useful URLs - Fibonacci Numbers and the Pascal Triangle, which will be referred to from time to time, will help beginners like myself further explore the golden ratio and Fibonacci Numbers.

*1 06/11/18
((phi + 1)^10) - (1 + (phi^10) + ((10 * phi) * ((phi^8) + 1)) + ((45 * (phi^2)) * ((phi^6) + 1)) + ((120 * (phi^3)) * ((phi^4) + 1)) + ((210 * (phi^4)) * ((phi^2) + 1)) + (252 * (phi^5))) = 0

*2 06/11/26
the golden ratio^20 = 15 126.9999

(phi^20) - ((1 + phi)^10) = 0

(phi^10) - sqrt((1 + phi)^10) = 0

the golden ratio^10 = 122.991869

the golden ratio^5 = 11.0901699


Excel Calendar

Exploring Phi-by-Phi Squares with Excel Trigonometry

((phi^3)^2) - ((phi^4) + (phi^5)) = 0 is an exmple of how phi^n makes the Fibonacci Sequence - phi^n=phi^n-2+phi^n-1.

Excel is used to show how multiplied phi-by-phi squares are analysed. The phi^2-by-phi^2 square has an area of phi^4.
Phi^4 is equal to: 1 + (4 * (1 / phi)) + (6 * (1 / (phi^2))) + (4 * (1 / (phi^3))) + (1 / (phi^4))). They can be broken down to smaller squares:

phi^3-by-phi^3 "Golden Square", 4*1/phi^2 squares, and 4*1/phi-by-(phi^2-1/2*phi) rectangles

Excel Trigonometry

Exploring Pi with Trigonometry

Trigonometry that seems to have nothing to do with circles or anything having no angles does work in measuring tangent pi on angles on triangles with an area equal to or twice as large as that of a circle and having angles with pi or 2*pi in tangent.

pi-tan(72.343212845degrees) - Google Search -

My Excel sheets show how calculation was made and the relevant trigonometric function.

Excel Trigonometry