# Math Photo: Lego-Linear Approximations

This replica of The Thinker at Legoland got me thinking about linear approximations.

One of the fundamental ideas in Calculus is that certain kinds of curves can be very closely approximated by straight lines. In fact, when examined closely enough, these differentiable curves are essentially indistinguishable from straight lines. This is important because lines are easy to understand and analyze, whereas curves can be very complicated.

We see this phenomenon at play in Lego sculpture. Here, The Thinker’s curves are being approximated by rectangular Legos, and beautifully so. And scale plays an important role: a larger Thinker looks better in Lego than a smaller one, because the finer the approximation, the better the fit. This is something that any child who has ever tried to make a Thinker out of handful of Legos knows this firsthand.

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#### Dan Anderson · May 20, 2018 at 2:13 pm

This is a great thought, and I love how you can use these kinds of thoughts to expand hs calculus students to multidimensional calculus. Here’s a desmos version of this idea (2D) that uses squares to approximate an implicit inequality. https://www.desmos.com/calculator/asc8ttabq3

#### MrHonner · May 20, 2018 at 7:04 pm

Yes, I love how these ideas can build simple, powerful bridges between single- and multi-variable Calculus.

I liked your second Desmos graph, too! Interested readers can find it here.

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