A Heart That Continually Grows Forever or Infinitely

By Sophia Zhanissov (grade 6) and Maria Droujkova

Some math activities are so rich in mathematical connections, inspirations, and energy! If mathematical routines were people, Growing Patterns would be a pillar of the community. Growing Patterns routine connects several key ideas in computer science, algebra, geometry, and beyond. Moreover, all that is doable at the pre-algebra level; you’ll be essentially learning algebraic reasoning from scratch. Here’s what we did over the hours of the project, in a nutshell:

1. Admired Fawn Nguyen’s (last name is pronounced “win”) whimsical collection of hundreds of growing patterns at VisualPatterns.org and decided to add to it!
2. Made our own visual growing patterns using graph paper, colored pens, and wooden cubes.
3. Using spreadsheets, scripted recursive formulas for a few starter number sequences: counting numbers, even numbers, odd numbers, and alternating numbers (1, -1, 1, -1…).
4. Scripted recursive formulas for the three patterns we’d like to add to the collection.
5. Wrote stories explaining the three patterns.
6. Scripted closed formulas for n-th entries in the three patterns.
7. Looked at the corresponding functions using a grapher (Desmos).

The drawings, stories, functions, formulas, and scripts are all different representations of the same infinite, growing sequences. They help us understand the mathematical nature of the sequence and express its growth in a single closed formula. That formula, for us, evokes many geometric, algebraic, and computer-science connections, kind of like e=mc^2 packs a vast amount of physics for those who know how to derive it.

A Very Long Pool

A very long pool is my pattern, where the water is some blue squares(or opposite), the Rim is the green squares making an equal shell around the water(or opposite once again), and the stairs is the singular green square on the right side of the pool. The “very long pool” is essentially a pool that grows by the second, making it (technically) infinite or just very long.

The number of squares and recursive formulas for the water, rim, stairs, and the total space of the pool:

Closed formula for the Nth total space: N*6+4.

Heart

Growing emotions is one of my patterns, it is a heart that continually grows forever or infinitly. The yellow squares inside is the inner heart, where it beats and grows in size, and the purple squares outside is the outer heart, that protects the inner heart from dangers. Given that is grows infinitly, it isn’t strong enough to be on it’s own, and is very sensitive(emotional) and prone to damage. Together they grow and grow to the point where there is no stopping point.

The number of squares and recursive formulas for the inner heart, the outer heart, and the total number of squares:

Closed formula for the Nth total space, outer heart: N*4+6; inner heart 2*N^2+4*N-3.

Road to… Nothing

The road to nothing is my pattern, where there is a road, a very long road, or you could just say, infinite. Most patterns go infinite if there is no stopping point, so does this pattern. This road leads to, you guessed it, nothing… If there is no stopping it, there is no end which means it leads to nothing. It may come across one or two things, but it doesn’t stop going, it leads no one, once again. It has no purpose, it is useless you may call it, it has no emotions, unlike the growing emotions pattern, so it has no feelings, so say whatever you want to it, as it won’t even respond. But anyways, the yellow part is some holes in the road, and the purple is some stepping pads that make up the road.

The number of squares and recursive formulas for the green parts, pink parts, and total squares in the road:

Note that in this example, the closed formula for the total number of squares in the road to nothing is quite easy. But alternating recursive patterns by color took some doing, the way we did them!

Closed formula for the Nth total bricks in the road: N*3.