Arts and Humanities Commons^™

One Theorem, Two Ways: A Case Study In Geometric Techniques, John B. Little

Journal of Humanistic Mathematics

If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in same ratios:

AH : HB = BΘ : ΘΓ = ΓK : KA,

then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic …

Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs.

One Straight Line Addresses Another Traveling In The Same Direction On An Infinite Plane, Daniel W. Galef

Journal of Humanistic Mathematics

No abstract provided.

Pattern Blocks Art, Gunhan Caglayan

Journal of Humanistic Mathematics

Pattern blocks are versatile manipulatives facilitating connections that can be made among various strands of mathematics such as number sense, algebra, geometry and measurement, spatial reasoning, probability and trigonometry. This note focuses on an artistic interpretation of the pattern blocks with primary focus on convex polygons made with pattern blocks, and describes five mathematically rich activities using them.

Raphael's School Of Athens: A Theorem In A Painting?, Robert Haas

Journal of Humanistic Mathematics

Raphael's famous painting The School of Athens includes a geometer, presumably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael's lifelong friendship with the eminent mathematician Paulus of Middelburg. It then presents several conjectured explanations, notably a theorem about a hexagram (Fichtner), or alternatively that the construction may be architecturally symbolic (Valtieri). The author finally offers his own "null hypothesis": that the scene does not show any actual mathematics, but simply the fascination, excitement, and joy of mathematicians at their work.

Logarithmic Spirals And Projective Geometry In M.C. Escher's "Path Of Life Iii", Heidi Burgiel, Matthew Salomone

Journal of Humanistic Mathematics

M.C. Escher's use of dilation symmetry in Path of Life III gives rise to a pattern of logarithmic spirals and an oddly ambiguous sense of depth.