## Introduction: How to Create Regular Tetrahedron in 3d CAD

This series will follow several different attempts to create regular tetrahedrons, a 3 sided pyramid with all edges being equal length. The first set of explorations will take place in Autodesk Inventor.

The tetrahedron is a rather interesting case of a shape which can be rendered in a cad application in several ways with differences in accuracy and complexity. by exploring these different ways of constructing a 3d model of this simple shape we can learn a great deal about how strengths of a given cad program and techniques and styles used in modeling. The tetrahedron, while conceptually simple can be a bit of a challenge to construct perfectly. Unlike a cube which is trivial to create perfectly the angles between faces of a tetrahedron are both compound and expressed by equations which result in irrational numbers. At the end of this exploration we will try approximations as well as ideal geometry proofs for construction of the tetrahedron.

Geometric perfection of this form is important when one enters in to stacking of tetrahedron as well as exploring the beautiful forms aperiodic tiling. This forms would have been extremely challenging to create in the past. Modern 3d printers enable physical exploration of these forms which until present have not been fully realized. These forms are only now being taken under study by most researchers despite being mentioned by historical figures such as Aristotle and Buckminsterfuller.

The tetrahedron is a rather interesting case of a shape which can be rendered in a cad application in several ways with differences in accuracy and complexity. by exploring these different ways of constructing a 3d model of this simple shape we can learn a great deal about how strengths of a given cad program and techniques and styles used in modeling. The tetrahedron, while conceptually simple can be a bit of a challenge to construct perfectly. Unlike a cube which is trivial to create perfectly the angles between faces of a tetrahedron are both compound and expressed by equations which result in irrational numbers. At the end of this exploration we will try approximations as well as ideal geometry proofs for construction of the tetrahedron.

Geometric perfection of this form is important when one enters in to stacking of tetrahedron as well as exploring the beautiful forms aperiodic tiling. This forms would have been extremely challenging to create in the past. Modern 3d printers enable physical exploration of these forms which until present have not been fully realized. These forms are only now being taken under study by most researchers despite being mentioned by historical figures such as Aristotle and Buckminsterfuller.

## Step 1: Creating a Base Equilateral Triangle in Autodesk Inventor.

This step we will create the base triangle of tetrahedron.

Open a new part in inventor. Select a plane and begin a new sketch. Draw a simple triangle and then add the constraint of equality to each side.

Tada! you just created an equilateral triangle!

you may also want to contrain the triangle with respect to the origin of the model. It is not nessisary though.

Open a new part in inventor. Select a plane and begin a new sketch. Draw a simple triangle and then add the constraint of equality to each side.

Tada! you just created an equilateral triangle!

you may also want to contrain the triangle with respect to the origin of the model. It is not nessisary though.

## Step 2: Tetrahedron

In the last step we created the base sketch. Now lets extrude the pyramid. In the extrude pop-up menu select the "more" tab. in the Taper field enter 70.5288-90 this is the approximate value of arccos(1/3)-90 which gives us draft angle for each face to create an approximate, though not exact tetrahedron.

Well done!

Well done!