No. 291

A new geometric design every day

No. 291

A new geometric design every day

Satie

**Inktober 2018 - Week 2**

Started getting into a stride on the second week, coming up with some interesting designs and trying new things. I really would like to spend more time learning to loosen up and make my designs appear more ‘organic’, as if they are breathing or being generated. I think this is achieved in some of these.

Late in the week however, I hit a bit of a creative wall. Feeling like a lot of what I am coming up with is too heavily inspired by others. Falling behind a bit on the Inktober schedule but I will be taking my time to make sure what I come up with feels like my own work.

*Hand drawn between Oct 8-15, 2018*

Geometric pattern BOU039

Orb

“Lunar”

cartographie de fouille

No. 279

A new geometric design every day

A beautiful geometric visualization of positive-valued **binomial expansions**, the algebraic expressions produced by raising sums of variables (*a, b*) to natural-valued powers n. The algebraic structures and procedures of binomial expansion are described by the **binomial theorem**–which, in turn, is proved by the above figures.

Given (a+b)^n, there will be (n+1)-many terms, c(*a*^(*n-m*))(*b^m*), where *c* is a constant, and *a* and *b* are variables.

By tradition, these terms are arranged in descending order by powers of the leading term, *a*. Accordingly, the binomial expansion of (*a+b*)^*n *begins with the term having the highest power of *a*, which is *a^n* for all *n*.

The remaining *n*-many terms are ordered such that the exponent of each successive *a* term (*n-m)* decreases by one. Correspondingly, the exponent of the *b* term (*m*) increases by one, such that (*n-m*)+*m=n*.

The binomial coefficients *c *for each successive term c(*a*^(*n-m*))(*b^m*) are described by **Pascal’s triangle**. Given (*a+b*)^*n*, the *n*th row of Pascal’s triangle contains (*n*+1)-many numbers, which are the coefficients *c,* listed in the order described above.

Note that the exponent *n* is the dimension of the figures pictured above. This is no coincidence; the term (*a+b*)^*n* can be depicted geometrically by a figure whose measure (length, area, volume, hypervolume, etc.) is the quantity produced by multiplying (*a+b*), *n*-many times.

Additionally, observe that the coefficients *c *give the quantity of *n*-dimensional figures required to fill in the missing bits of area, as illustrated by the figures. For example, in order to fill the volume (*a+b*)^3, we use two cubes of volume a^3 and b^3, and the remainder is filled by 3 “plate-like” figures and 3 “tube-like” figures (each of whose dimensions are (a^2)b and a(b^2), respectively).

Mathematics is beautiful. <3

I’m Back

Photographer: Fabrizio Raschetti

No. 240

A new geometric design every day

Eckhard Neumann, *Functional Graphic Design in the 20’s*, Reinhold, New York, NY, 1967

(via Letterform Archive)

By Jumbo Tsui for Sans Titre