GATE ME 2026 Set 1 — Question 45
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Metrology, CIM & Additive Mfg → Computer Integrated Manufacturing → CAD/CAM & Integration
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Question
Consider the Euler variables of polyhedral objects namely, , , , and as given in the table below.
Which one of the following options is NOT a topologically valid closed polyhedral object as per Euler’s law?
| Polyhedral objects | Faces () | Edges () | Vertices () | Faces' inner loops () | Bodies () | Genus () |
|---|---|---|---|---|---|---|
| 6 | 12 | 8 | 0 | 1 | 0 | |
| 5 | 8 | 5 | 0 | 1 | 0 | |
| 5 | 12 | 8 | 0 | 1 | 0 | |
| 10 | 24 | 16 | 0 | 1 | 0 |
Correct answer
(A) P3
Solution
Euler's law for a topologically valid closed polyhedral object is given by the formula: where is the number of vertices, is the number of edges, is the number of faces, is the number of bodies, is the genus (number of holes), and is the number of inner loops on faces. Let's check each object:
- For : ; . (Valid)
- For : ; . (Valid)
- For : ; . Since , is NOT valid.
- For : ; . (Valid)
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