3D Printing Mathematical Models

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3D Printing Mathematical Models – An easy way to visualize and understand mathematics

Introduction

We’re all familiar with how young children are taught basic geometry by playing with colorful toys in the shape of squares, triangles, and cylinders.

3D printing is being used to take that idea to the next level by creating physical models of highly complicated mathematical concepts. 

Physical models are a unique way to better understand complicated mathematical concepts and they can help students and mathematicians get to grips with concepts in a way that is impossible in any other medium. 

3D printing can be used to produce highly complex models that would be impossible to make by traditional means. 

In this article, I will explain why mathematical concepts are important and how 3D printed mathematical models can help us understand them.

Then I will elaborate on the benefits of 3D printing mathematical models, give you a few examples and show you how you can create your own 3D printed mathematical models. 

Mathematics and 3D Mathematical Models

Mathematics is all around us and mathematical concepts are used in every field of life, from natural sciences (chemistry, physics, biology) to engineering (mechanical, electrical) and in social sciences (political science, psychology, sociology, economics) as well.

Mathematical concepts are used to represent how things work in the real world. They help us to describe a system, to observe the properties of different components, and to predict their behavior. They provide us with solutions to our problems.

3D printers can bring these mathematical concepts to life. Taking them from our two-dimensional textbooks and chalkboards to the real three-dimensional world in the form of physical models. 

Benefits of 3D Printing Mathematical Models

There are many ways in which 3D printing mathematical models can help us. 

From basic mathematical concepts for school students to complex advanced mathematical concepts for post-graduate students, they can all be easily visualized with the help of 3D printed mathematical models. 

3D printing mathematical models can help in educating mathematics to students by making maths easy, interesting, and engaging. 

They can make concepts of calculus, geometry, and topology fun to learn. 

Visualization 

Students often find mathematics hard to grasp because they think it’s not applicable to everyday life. It’s hard to understand how all these abstract equations and proofs are related to the world around us. 

Being able to visualize these concepts will make it easier for students to know how mathematical equations and expressions look in real life.

Visualizing these concepts through models will aid in the better transformation of mathematical ideas to the students and they can utilize these concepts easily. 

3D mathematical models will play a role of a bridge between concepts in the books and the real world.

Here, you can see how parametric curves and volume of equations looks like with the help of a 3D printer. 

Parametric curves – Credit: Elizabeth Denne
Volume of defining equations – Credit: Elizabeth Denne

With the help of 3D printed mathematical models, you can see that all mathematical concepts and proofs are not just numbers and expressions but they are real-life objects. 

Hands-on Experience

Mathematical models are the only way to define geometric concepts such as surface, volume, width, symmetry, rotation, translation, area, edges, vertices, and faces, etc. 

Usually, the best way to learn basic geometry is to draw three-dimensional objects like cubes, spheres or cylinders on two-dimensional paper or board.

Drawing three-dimensional objects on two-dimensional objects and learning about their properties through the 2D (two-dimensional) diagram is a hard and confusing job for new students.

With the help of a 3D printer, you can have these models in your hands rather than on paper. This way, you will have hands-on experience as there is no alternative to that in learning. 

You can better understand their properties. You can compare them with the other models. And they will save you time and confusion.

As you can see here, the difference between a 2D diagram and it’s 3D printed model. 

2D Diagram – Credit: Kenneth Baker
3D Printed Model – Credit: Kenneth Baker

Having 3D printed mathematical models in your hands, you can examine them from every side and angle. You can make a prediction about their behavior and you can perform different experiments on them like rotation, or changing parametric values. 

You will see how giving different values to parameters will affect the shape of the object without any error.

Interest Development

To develop new technology or to advance existing technology, you must have a very good grip on mathematical concepts as they are the brain of technology. 

Technology works on the instructions given to it and these instructions are based on mathematical concepts.

So, it is very important to develop an interest in the new generation about mathematics if you want to advance in technology.

3D printing mathematical models can help to build the interest of the students to advance in mathematics by making it easy for them to understand the fundamental and complex mathematical concepts. This will lead to more innovations in both mathematics and technology. 

Below is an example of how making your own 3D mathematical objects can make you investigate new things.

Credit: Kelly Hong

“I’m always amazed at how quickly students learn to use technology tools,” says Hong. “Allowing students to use technology to solve problems, do research and present ideas gives them a better real-world understanding of the skills they’re developing, and it keeps them super-engaged.”

Because when you can actually transform something you’ve worked on to life and hold in your hands, it gives you a reason to keep learning.

Learning through irregular shapes rather than old, standard shapes in mathematics gives more realist experience.

You can create your custom models and can perform different measurements. It will enhance your knowledge.

Advanced 3D Printed Mathematical Models

Visualizing a mathematical object in 4D (four-dimension) or higher dimension without 3D printing them is virtually impossible.

Suppose visualizing the area of intersection of four cylinders or a cube in 4D by drawing them on a paper or a board. It will be a mess to draw them on a 2D paper or a board. We left with no choice but to imagine in our minds what they will look like.

With the help of a 3D printer, we can make models of any mathematical object, equation, or surface in 3D or higher dimensions. We can visualize those spaces about which we only see in mathematics.

Many mathematicians have already used 3D printers to visualize the concepts of geometry, calculus, and topology in four or higher dimensions. Henry Segerman is one of them. He has written a whole book on 3D printed mathematical models titled “Visualising Mathematics with 3D Printing”.

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In his book, Henry Segerman has discussed modern geometry, hyperbolic surfaces, knots, symmetry, four-dimensional shapes, and bunches of other concepts. These ideas were hard to envision. Images in his book consist of his own created mathematical models in 3D and in higher dimensions. 

His book emphases using 3D objects to visualize complex mathematical concepts along with commands for 3D-printing them and interactive simulations. It makes his book different from other books.

Below are the examples of a few mathematical models made by a 3D printer in higher dimensions. 

Examples of  Advanced 3D Printed Mathematical Models

3D printed mathematical models represent complex and advanced mathematics in a new, interesting, and engaging way. The facility to discover these objects from multiple angles is key to understanding them.

They can make conceptualizing easy for students and can unfold shapes, spaces and many theories in mathematics which were undiscovered till now.

Here, I have listed a few examples of 3D printed mathematical models that represent advanced and complex concepts of mathematics.

You will see how representing these concepts through 3D printed models can aid in building interest in mathematics in the students and they are just beautiful as a piece of art.

Fractal

Credit: George W. Hart

This is a picture of the 3D model of the popular fractal also known as “Sierpinski Tetrahedron”. It is a generalized form of a 2D Sierpinski triangle. It has five different sizes of octahedral holes which make it “fifth-version”.This model has been scaled such that the length of the edge of the tetrahedron is 8.5 inches. 

Round Möbius strip

Credit: Saul Schleimer, Henry Segerman 

A curve in space is defined by the borderline of a paper Möbius strip. That curve is not actually a circle, but it can be turned into one. The following sculpture shows a large version of the round Möbius strip. It is the result of dragging the surface of the Möbius strip as you “flatten out” the circle. One point has been dragged off the strip to infinity for the sake of symmetries of the Möbius strip and round circle is visible at the center of the model. It represents the boundary line of the original Möbius strip.

3D Printed Math Art

3D printed mathematical shapes don’t have to be understood to be useful. They can also be appreciated just for their aesthetic value. 

This tenet has led to the development of math art which uses mathematical geometries as the foundation of art pieces including sculptures and jewelry. 

Check out the excellent designer Luxxeon3D on Shapeways for an example of math art being used in this way. 

-Dylan

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