The towering structure in the above photo is the result of a remarkable collaboration between the Allegany-Limestone, Hinsdale, Olean, and Portville school districts called The Connect 4 Project. The 13-foot tall, 32,770-piece structure was built by 87 middle school students using a construction system called Zometool. Mathematically, the structure represents a beautiful geometric figure called a stage-6 Sierpinski tetrahedron (or stage-6 tetrix). The Connect 4 Project was organized by Chris Hill, Assistant Professor of Mathematics at St. Bonaventure. The project was generously supported by Cattaraugus-Allegany BOCES.
In each of the four districts, a group of students was guided by their teachers to build a stage-5 Sierpinski tetrahedron, which stands over six feet tall and contains 8,194 Zometool pieces. Each teacher or group of teachers devised a schedule for building their stage-5 structure that worked best for their students, with some groups completing their portion in the fall of 2014 and others completing theirs in the spring of 2015. Linda Dodd-Nagel worked with 19 eighth-graders at Allegany-Limestone Central School; Holly Edwards and Matthew Taub worked with 36 fifth-graders at Hinsdale Central School; Jake Palo, Christine Armstrong-Gabler, Maureen Ullman, and Katie Ralston worked with 18 students at Olean Intermediate Middle School; and Bill Torrey and Matt Burlingame worked with 14 seventh- and eighth-graders at Portville Central School.
Many of the students, teachers, and administrators involved in the project were present at the grand finale, which was held on May 7, 2015 in the Quick Center for the Arts' Rigas Family Theater. At the finale, the four stage-5 structures were connected to make a stage-6 Sierpinski tetrahedron. With the theme music to "2001: A Space Odyssey" playing over the stage, a group of teachers gently placed one stage-5 on top of the other three to complete the final structure. And... the structure held! The following video captures the completion of the Connect 4 Project. In the words of Helen Keller: Alone we can do so little. Together we can do so much.
After the world premier at the Quick Center, the Zometool stage-6 Sierpinski tetrahedron went on tour, spending time at Olean Intermediate Middle School, Allegany-Limestone Central School (for their Arts Festival), Hinsdale Central School (for their Spring Concert), Portville Central School, and finally the Olean Center Mall.
We believe so! In a congratulatory letter to the students and teachers, the Chief Visionary Officer of Zometool, Paul Hildebrandt, stated that, to his knowledge, this was the first time a stage-6 Sierpinski tetrahedron had ever been built using Zometool. Zometool President Carlos Neumann featured the project on the company’s main website and facebook page.
This endeavor, which spanned two years from conception to completion, was called The Connect 4 Project for two reasons. First, the four school districts involved are those in the Connect 4 Program, which facilitates the sharing of resources among the four districts. Second, the four stage-5 structures were literally connected to form the stage-6 structure.
The world's first stage-6 Sierpinski tetrahedron made from Zometool was built by 87 middle school students from Allegany-Limestone Central School, Hinsdale Central School, Olean Intermediate Middle School, and Portville Central School. In the following photos, each group of students poses with their completed stage-5 structure or works on their stage-5 structure in their own school.
The Allegany-Limestone group, guided by Linda Dodd-Nagel.
The Hinsdale group (consisting of two classes), guided by Holly Edwards and Matthew Taub..
The Olean group, guided by Jake Palo, Christine Armstrong-Gabler,
Maureen Ullman, and Katie Ralston.
The Portville group, guided by Bill Torrey and Matt Burlingame.
One student from each school was selected to represent their group in a
special photographic memento of the occasion, as seen above and in the
close-up below. Go Bobcats, Huskies, Panthers, and Gators!
With powerful theater spotlights illuminating the structure from
above, the Sierpinski tetrahedron practically glows.
In the image above, the theater lights were off and a flash, placed
behind the structure, was triggered at the instant the photo was
taken. Check out the cool shadow on the floor.
With the flash in front of the structure and a white screen behind, we got another awesome shadow.
Dr. Chris Hill hangs out within a stage-6 Sierpinski tetrahedron.
The weather inside is really nice.
A Sierpinski tetrahedron or tetrix is a certain geometric figure. It has the shape of a tetrahedron which is comprised of four smaller tetrahedra, each of which is comprised of four even smaller tetrahedra, each of which is comprised of four even smaller tetrahedra, and so on, ad infinitum. A Sierpinski tetrahedron is an example of a fractal. The figure is named for the Polish mathematician Wacław Sierpiński, who studied the two-dimensional analogue called a Sierpinski triangle.
A Sierpinski tetrahedron contains infinitely many “levels” of smaller tetrahedra. Since it contains an infinite amount of detail, a Sierpinski tetrahedron cannot be physically built. However, approximate models of a Sierpinski tetrahedron can be built. The "stage" of an approximate model indicates the number of levels of smaller tetrahedra in the model. A stage-0 Sierpinski tetrahedron is simply a tetrahedron—it contains no smaller tetrahedra. A stage-6 Sierpinski tetrahedron contains six “levels” of smaller tetrahedra. The image to the right shows a model of a stage-5 Sierpinski tetrahedron that was 3D-printed by George Hart.
Zometool is a mathematically-precise plastic construction system for building geometric structures, from simple polygons to “shadows” of four-dimensional figures, from models of DNA molecules to works of art.
A Zometool model of a particular stage of a Sierpinski tetrahedron may be built using connector balls and either red struts and blue struts or green struts. We chose to use reds and blues to make the structure more visually interesting. Also, since the connector balls are white, the red and blue struts make for a patriotic project! The one disadvantage to using red struts and blue struts (rather than green struts) is that the red struts and blue struts do not have exactly the same length (although they're close), so the resulting tetrahedra are not regular (although they're close). The photo to the left shows, from left to right, a stage-0, a stage-1, and a stage-2 Sierpinski tetrahedron made from Zometool.
By making four copies of a particular stage of a Sierpinski tetrahedron and connecting them in a tetrahedral arrangement, you build the next stage of a Sierpinski tetrahedron. For this reason, the stage-6 Sierpinski tetrahedron was perfectly-suited for a project built by four groups of students.
A Zometool stage-6 Sierpinski tetrahedron with red and blue struts requires 8194 balls, 12288 R1s, and 12288 B1s, for a total of 32770 parts. In general, for any positive integer n, a Zometool stage-n Sierpinski tetrahedron requires 2(4n) + 2 balls, 3(4n) R1s, and 3(4n) B1s, for a total of 8(4n) + 2 parts
The Connect 4 Project was generously supported by Cattaraugus-Allegany BOCES.
The project would not have been possible without the support and guidance of the administrators from the four districts: Superintendent Karen Geelan of the Allegany-Limestone School District; Principal Cory Pecorella of Allegany-Limestone Middle School; Superintendent Larry Ljungberg of the Hinsdale School District; Principal Laurie Cuddy of Hinsdale Central School; Superintendent Colleen Taggerty of the Olean School District;
Principals Joel Whitcher (grades 4, 5), Gerald Trietley (grades 6, 7), and Barbara Lias (grades 8-12) of Olean Intermediate Middle School and High School; Superintendent Tom Simon of the Portville School District; and Principal Larry Welty of Portville Central School.
Ludwig Brunner, Executive Director of the Quick Center for the Arts, provided the Rigas Family Theater, which was the perfect venue for the finale. Don Hopwood, Technical Director of the Quick Center, gave us indispensable assistance in staging the finale.
Clara Wilder of the Olean School District's Central Receiving transported the structure for every leg of its tour, which made the tour possible.
For more about mathematics outreach at St. Bonaventure, please contact Dr. Chris Hill at firstname.lastname@example.org or (716) 375-2025. To learn more about Zometool, see Dr. Hill's page of Zometool Resources.
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