In 2023, I got to build what I call a math teacher’s geometric dream house. It’s in the form of a rhombic triacontahedron, a form that my 8th grade students might recognize from the geometry block in which we study Platonic and Archimedean solids. The Waldorf geometry curriculum is one of the many ways that we develop the imagination of the students. The 8th grade geometry curriculum allows the students (through working with both paper and clay) to experience the metamorphosis of form.
I have called my geometric dream house the Phi House because the Golden Ratio (Φ, Phi) is found in multiple places within the rhombic triacontahedron. Some explanation may be helpful. The Golden Ratio (Φ) comes from the pentagon. It answers the question: How many times longer is the diagonal (D) than the side (X)? (See the drawing on the right.) The answer is that the diagonal of a (regular) pentagon is Φ times longer than its side, where Φ is equal to approximately 1.618.
The rhombic triacontahedron consists of thirty identical rhombuses (diamonds). One of the places that Φ appears in this form is with each of the rhombic faces. As shown in the drawing here, the long diagonal of each rhombus is Φ times longer than the short diagonal.
You may wonder why I chose to build my house in the form of a rhombic triacontahedron. I have a special fascination with the pentagon (which is the foundation of my house), and its three-dimensional equivalent: the (pentagonal) dodecahedron (which is shown on the right). It turns out that there is only one form that comes out of the dodecahedron, which has the possibility of vertical walls if you are building a house – and that’s the rhombic triacontahedron.
In the 8th grade geometry block I taught for many years, wide-eyed students would observe as I pulled one wondrous, colorful geometric shape after another out of my black, somewhat battered, old-fashioned trunk. I felt like a magician lining up his props for prestidigitation. The students and I named each shape and described its qualities. Eventually, we arrived at my favorite, the rhombic triacontahedron. At that point, I would declare my desire to someday build a house with this form; it would be such a feat, to create this work of architectural work of art, that to my knowledge, no-one had ever accomplished before. The Phi house sits on five stilts, its three oversized rhombic windows look out on a nearby mountain range, and its skylights invite cosmic observation at night.
In life, it is quite something when we can manifest a dream. For me, the construction process brought me the greatest joy; it was more about the journey than the destination, although the finished result is rather spectacular, especially when its metallic “Dragon Armor” clad exterior glints varying colors depending on the vantage point from which it is viewed.
I’ve shared the video I made about this journey with former and current students. I’ve wanted to inspire them to follow their dreams in life and to visit the Phi House if their travels take them to the Western Slope of Colorado and the small valley town of Paonia. That invitation goes out to others too, who are curious to see this mathematician’s wild geometric dream house.
I’ll leave you with a link to this short video about the construction process: https://www.youtube.com/watch?v=TCbXNHgrBHY
Feel free to email me with questions at [email protected]. Videos about my approach to math education are also available on our YouTube channel: https://www.youtube.com/@waldorfmathematics
Jamie York
Jamie York is an internationally respected Waldorf math consultant, and lead author of Making Math Meaningful™ books. Jamie was born in Maine, went to public school in Connecticut, received two computer science degrees, and then began teaching math in 1985 at a boarding school in New Hampshire. In 1994, after spending two years in Nepal serving as a Peace Corps volunteer, Jamie’s search for meaningful education led him to Shining Mountain Waldorf School (in Boulder, Colorado), where he taught middle school and high school math for 21 years. Now, as a math missionary, he travels across the U.S. and internationally offering workshops for teachers so they can re-imagine mathematics and inspire their students. Making Math Meaningful™ books are the manifestation of his vision for a comprehensive mathematics curriculum that spans grades one through twelve. Jamie’s website, online math workshops, and books are resources for Waldorf teachers around the world.
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