About

The Euler characteristic is a number χ that describes the overall 3D shape regardless of how it is deformed. This number is calculated by χ=V−E+F where V is the number of vertices, E is the number of shapes and F is the number of faces. In its simplest form, this number can be expressed as χ=2-2T, where T is the number of tunnels. Thus, a cube and a sphere have both χ=2, while a torus has χ=0, because T=1. Another way to calculate the Euler characteristic is by the Euler-Poincaré formula.

Click on the image to start the application.

Alternatively, here are versions in English, Bulgarian and Japanese.

How to play

When the application is started it shows a grill with a small cube. When a challenge is started this cube grows into a randomly generated 3D polyhedron. Its Euler characteristic must be calculated. The level of difficulty determins how complex the object is and how fast the grill is spinning it. The goal is to reach score 100.

Here are a few hints: Using χ=V−E+F might be cumbersom, and it is applicable only to simple 3D shapes. An easier approach is to use χ=2-2T, where the difficulty is to find the number of tunels T. The reason for this difficulty is that topological tunnels are not the same as our everyday interpretation of a ‘tunnel’:



The following 3D examples demontrate a few objects generated by the application. The first object has no tunnels, so χ=0. The second object has χ=4, because it is composed of 2 separate objects, each having χ=2. The third object has one tunnel, so χ=0. And the last object has χ=-4, because there are three tunnels.

Integration with LMS

This application is provided as SCORM (Sharable Content Object Reference Model) module. It can be used with any LMS (Learning Management System) that supports version SCORM 1.2. SCORM modules are delivered as ZIP archive.

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Follow the instruction of your LMS on how to install a SCORM module. Usually the ZIP is uploaded and a few additional settings are set.

When run from a LMS, the application reads these data:

When run from a LMS, the application sends back these data:

Data policy

The application itself does not create or use HTTP cookies, web beacons, spy pixels or any other tracking technology. Besides SCORM-related data, described in section Integration with LMS, the application creates a local storage entry called 'sound' with values 'on', 'off' or 'fx'. This entry is used to record user’s sound preference and it is not sent to the server.

When the application is run from a LMS, the LMS may utilize its own data policy, which is beyond the scope and the control of this application.

Disclaimer

The application shows constantly spinning object that is the attraction focus. This may cause dizziness in people prone to vertigo.

Credits

This application uses a background music from SoundCloud:

and sound effects from Mixkit:

November, 2022