Learning mathematics and its applications to real life can be challenging tedious. This project uses 3D printing to demonstrate wave reflection properties of parabolas, hyperbolas and ellipses. The principles learned here can be applied to various aspects of engineering such as RADAR engineering, radio transmission and reception and lens manufacturing.
Having taken Algebra 2 level mathematics in school, I understand the reflective and refractive properties of conic sections like parabolas, hyperbolas and ellipses. I realized it would be possible to utilize commonly owned smartphones (like iPhones) to be a source of the sound waves to demonstrate the reflective properties of such shapes.
I want students to have a fundamental understanding of the properties of conic sections, so each smartphone attachment has an equation the student can graph and analyze.
I used the native Mac OS X grapher tool to test out equations which I would then model in 3D.
This image shows the basic principle behind a parabolic reflector. The sound waves are emitted from the focal point, hit the parabola, and due to the geometry of a parabola are reflected parallel to each other, perpendicular to the vertex plane. Source: www.stagelightingprimer.com
Designing a hyperbola
3. Designing the iPhone Adapters
Since iPhones are so common in the modern classroom, I figured I would use them to my advantage. I designed three adapters that would work on the iPhone 6 Plus, 6S Plus, 6, 6s, 5 and 5S.
I used the free and open-source CAD software "FreeCAD" to design the adaptors.
4. Model the Conic Sections
I used Blender to model each equation, starting from a hemisphere and scaling each row of vertices. I attached the models of each graph to a connector piece I designed in FreeCAD.
Hyperbolic Sound Director
Parabolic Sound Director
Model With Connector Piece (Cylinder at the bottom)
5. Boolean the Equations to the Side of the Connector Piece
Blender has a native text-to-mesh generator, so I used that to create 3D models of the equations.
Model of Equation
Connector Piece with Equation
Learning about the reflective properties of 3D Parabolas, Hyperbolas, and Ellipses.
Overview and Background:
Parabolas, Hyperbolas and Ellipses, otherwise known as conic sections, are fundamental geometries in engineering, science, and mathematics. Parabolas can predict where a thrown baseball will land; hyperbolas can determine the shape of rope hanging by its two ends; ellipses can describe the motion of planets. These concepts are easily taught in a classroom environment because they are rather intuitive.
What might be difficult for a child to understand are the reflective properties of conic sections. While it may seem trivial to a student, these properties allow for remarkable feats of engineering like RADAR and radio reception.
This project allows students to experiment with various conic section-based geometries. A student can experiment with the different 3D printed attachments, and using their smartphones, can generate all types of sound using free sine wave generators from the App Store, or even their own music! By hearing the variations in the volume of the sound after being reflected, students can make their own conclusions, confirmed by the teacher, as to the properties of the conic sections.
A student will learn the following from the project, with appropriate instruction from the teacher.
- How various conic sections reflect sound waves
- The design characteristics of parabolas
- The design characteristics of hyperbolas
- Engineering applications of conic sections
- The mathematical representations of 3D rotated conic sections
- The fundamental relationship between the focal point of the conic section and the graph itself
- The fundamental differences between 2D and 3D conic sections
6th grade - 10th grade (10 - 15 years old)
I designed this project to be fun and informative for a wide age group. Younger audiences do not have to learn the mathematical representations of the conic sections, nor detailed descriptions of conic sections. They can simply experiment with various sounds and reflectors, figuring out the differences between the various geometries. The teacher can then give general explanations for why sound is reflected the way it is. (see "references" subsection)
The lesson plan (below) is designed for a high school audience.
The principles learned here are applicable to mathematics, physics, and engineering.
The project would ideally be utilized in an Algebra 2 level (or higher) math class during the conic sections unit.
Students who complete this project will have a better understanding of mathematics when it comes to engineering applications. They will also understand the fundamentals behind so many modern technologies that rely on such shapes.
There are three types of 3D printed materials that the students and teacher should know of:
iPhone adapters -- Connect these to the iPhone at the loudspeaker output
Reflectors -- Attachments which connect to the iPhone adapters; these reflect the sound waves in a completely new direction.
Directors -- Attachments which connect to the iPhone adapters; these "direct" sound waves to a singular path, without drastically changing the direction of the waves.
See videos below for a demonstration.
- Background information for the students
Students of all ages would require some background information on conic sections. At the high school level, the teacher would go over the section on conics in a math textbook. This would entail going over base functions of a parabola, hyperbola and ellipse, and transformations of each. Ideally, this project would be in use during the conic sections unit of an Algebra 2 class.
More information will be found in the "references" subsection.
- Take out iPhones
The project is designed to work with the iPhones 6 Plus, 6S Plus, 6, 6S, 5, 5S and SE. The student can use any song or video for sound, or they can download a sine wave generator from the App Store and experiment with sounds of varying frequency. The students should select their appropriate adapters.
The student can install the adapter, and can test out each of the reflectors/directors. They should note down which sound reflector/director they are using, and discuss what they observe.
Key things to observe are:
- Where is the sound the loudest? Is it the same volume all around the smartphone?
- What happens if you move the attachment a few centimeters from the connection point (still in the direction of the speaker output)? Does the sound still reflect the same? Why does it or why does it not?
- Does the loudness (magnitude) or frequency of the output sound matter when it comes to reflection of the sound waves?
- How does the quality of the reflected sound change in relation to the loudness?
The student should draw cross sectional diagrams tracing lines (representing sound waves) from the source to after they have been reflected.
- Analyze the graphs
This is the mathematical part. This will part will allow the students to discover for themselves why the reflectors/directors work the way they do.
Every conic section has one or more focal points. In order for the reflectors to properly work, the output sound must come directly from one focal point. (see image below)
The teacher should ask each student to graph (using a graphing calculator or program) the equation that is on each reflector/director, identify the focal point, and try to figure out whether or not the sound output is directly at the focal point of each 3D printed model (Every model except the Parabolic Director #2, Hyperbolic Reflector #2, Hyperbolic Director should have the focal point at the sound output point). Formulas for finding focal points can be found in any Algebra 2 textbook, or in the "references" subsection located below.
The teacher should tell the student that the equations for 3D graphs (equations with x, y AND z) can be converted to a 2D cross sectional graph.
- Teacher confirms the results of students
Each student should have come up with her/his own notes on each of the reflectors. The teacher can then display the image (below) and answer any questions. The math teacher can also explain the importance of focal points.
See "references" subsection for more information regarding focal points.
- (For calculus based courses only)
If the students know basic calculus, the teacher can walk through the derivatives of each equation, and by analyzing the angle of incidence and angle of reflection with respect to the focal point and a point on the graph, he/she can draw an accurate diagram (similar to the image below) for the reflection of sound waves for each particular conic section. Such a diagram would exactly match the cross section of each 3D printed reflector/director, and would exactly represent the sound waves traveling from the source to the reflector.
This is a good visual aid to hand the students after they are done experimenting. It can help students visualize why the sound output must be at the focal point of the graph. It can also be used as a reference for the teacher before the lesson.
An iPhone 6 Plus with the adapter. The output point it labeled.
Each student puts together the components like this.
Demonstration of the kit using music. The video has annotations.
Demonstration of the kit using a free tone generator from the App Store. The video has annotations.
This should take up one class period; roughly 50 minutes. This is enough time for the teacher to cover the basics of conics, to allow the students to experiment with the various reflectors and directors, jotting down notes and diagrams as they go along, and to allow the teacher to finish off the experiment with a brief confirmation/explanation of the student's findings (details in the "references" section).
- 3D printer
- Algebra 2 level textbook (or websites listed in "references" subsection)
- iPhones with music or YouTube access
- Access to the App Store (optional, for free tone generator apps)
- Graphing calculator/mobile app/software
It is assumed any teacher teaching the high school level lesson plan already has a background in STEM and therefore can answer specific questions the students may have.
Rubric & Assessment
Since there are many ways a teacher can use this project, the grading should be up to the teacher. The teacher can grade the notes taken (as part of the lesson plan).
By the end of the project, the student should have an understanding of the reflective properties of conic sections, as well as the mathematical representations of 3D conic sections.
Handout & Assets
Since this project can be used for a wide range of ages, I believe it would be best for the students to take class-specific handwritten notes.
I am a 16 year old high school student. I live in Silicon Valley.