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Mathematical Curiosities

                                   More number patterns

  With 9s we can produce another aesthetically pleasing pattern shown below. 

                                         999,999 · 1 = 0,999,999

                                         999,999 · 2 = 1,999,998

                                         999,999 · 3 = 2,999,997

                                         999,999 · 4 = 3,999,996

                                         999,999 · 5 = 4,999,995   

                                         999,999 · 6 = 5,999,994

                                         999,999 · 7 = 6,999,993

                                         999,999 · 8 = 7,999,992

                                         999,999 · 9 = 8,999,991

                                         999,999 · 10 = 9,999,990

Here is another number pattern, where the number 9 is multiplied by a number representing the consecutive natural numbers- increasing by 1 each time- and then added to the initial natural numbers consecutively. 

                                                   0 · 9 + 1 = 1

                                                   1 · 9 + 2 = 11

                                                 12 · 9 + 3 = 111

                                               123 · 9 + 4 = 1111

                                             1234 · 9 + 5 = 11111

                                           12345 · 9 + 6 = 111111

                                         123456 · 9 + 7 = 1111111

                                       1234567 · 9 + 8 = 11111111

                                     12345678 · 9 + 9 = 111111111

We can consider number patterns that can be generated in a similar fashion, yet somewhat in the reverse of the previous one. However, this time the generated number consist of all 8s. 

                                                  0 · 9 + 8 = 8

                                                   9 · 9 + 7 = 88

                                                 98 · 9 + 6 = 888

                                               987 · 9 + 5 = 8888

                                             9876 · 9 + 4 = 88888

                                           98765 · 9 + 3 = 888888

                                         987654 · 9 + 2 = 8888888

                                       9876543 · 9 + 1 = 88888888

                                     98765432 · 9 + 0 = 888888888

Now that we have introduced the 8s in a rather dramatic fashion, we shall use them as the multiplier with the numbers consisting of increasing natural numbers, and each time adding successive natural numbers. Appreciating the number pattern shown here is more pleasing than trying to explain this phenomenon, which could conceivably detract from its beauty. 

                                                  1 · 8 + 1 = 9

                                                12 · 8 + 2 = 98

                                              123 · 8 + 3 = 987

                                            1234 · 8 + 4 = 9876

                                          12345 · 8 + 5 = 98765

                                        123456 · 8 + 6 = 987654

                                      1234567 · 8 + 7 = 9876543

                                    12345678 · 8 + 8 = 98765432

                                  123456789 · 8 + 9 = 987654321

Content credit: Mathematical Curiosities A Treasure Trove of Unexpected Entertainment 

LEGO Nautilus, Ammonite, and Trilobites

Created by Tim Goddard

The fittest fit girls

Mood

A short note on how to interpret Fourier Series animations

When one searches for Fourier series animations online, these amazing gifs are what they stumble upon.

They are absolutely remarkable to look at. But what are the circles actually doing here?

Vector Addition

Your objective is to represent a square wave by combining many sine waves. As you know, the trajectory traced by a particle moving along a circle is a sinusoid:

This kind of looks like a square wave but we can do better by adding another harmonic.

We note that the position of the particle in the two harmonics can be represented as a vector that constantly changes with time like so:

And being vector quantities, instead of representing them separately, we can add them by the rules of vector addition and represent them a single entity i.e:

                                                  Source

The trajectory traced by the resultant of these vectors gives us our waveform. 

And as promised by the Fourier series, adding in more and more harmonics reduces the error in the waveform obtained.

        Have a good one!

**More amazing Fourier series gifs can be found here.

Some sceneries and living spaces.

I wanted to get some ideas/feelings out of head and to try out some new techniques.

Full-resolutions:  1    2    3

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