Example 2.6
To use the Euler–Poincare Law to justify the validity of open objects in the first column of Table 2.8.
Table 2.8 Examples of open objects.
| Example | F | E | V | L | B | G | F − E + V − L = B − G |
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0 | 1 | 2 | 0 | 1 | 0 | 0 − 1 + 2–0 ≡ 1 − 0 |
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0 | 3 | 4 | 0 | 1 | 0 | 0 − 3 + 4 − 0 ≡ 1 − 0 |
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2 | 8 | 8 | 1 | 1 | 0 | 2 − 8 + 8 − 1 ≡ 1 − 0 |
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2 | 2 | 2 | 1 | 1 | 0 | 2 − 2 + 2 − 1 ≡ 1 − 0 |
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1 | 4 | 4 | 0 | 1 | 0 | 1 − 4 + 4 − 0 ≡ 1 − 0 |
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5 | 12 | 8 | 0 | 1 | 0 | 5 − 12 + 8 − 0 ≡ 1 − 0 |
Solution
As shown in the middle six columns of Table 2.8, the number of faces (F), edges (E), vertices (V), bodies (B), inner loops on faces (L), and genuses (G) in a geometry is counted and the valid condition for a simple solid by the Euler–Poincare Law is applied. It shows that all of the objects satisfy the conditions as open objects.
Example 2.7
To use the Euler–Poincare law to justify the validity of generic objects in the first column of Table 2.9.
Table 2.9 Examples of generic objects.
| Example | F | E | V | L | B | G | F − E + V − L = 2(B − G) |
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13 | 27 | 18 | 2 | 1 | 0 | 13 − 27 + 18 − 2 ≡ 2(1 − 0) |
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10 | 24 | 16 | 2 | 1 | 1 | 10 − 24 + 16 − 2 ≡ 2(1 − 1) |
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7 | 13 | 10 | 0 | 2 | 0 | 7 − 13 + 10 − 0 ≡ 2(2 − 0) |
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12 | 24 | 16 | 0 | 2 | 0 | 12 − 24 + 16 − 0 ≡ 2(2 − 0) |
Solution
As shown in the middle six columns of Table 2.9, the numbers of faces (F), edges (E), vertices (V), bodies (B), inner loops on faces (L), and genuses (G) in a geometry are counted and the valid condition for a simple solid using the Euler–Poincare Law is applied. It shows that all generic objects satisfy the conditions as generic objects.
2.3.3 Euler–Poincare Law for Solids
Objects in the physical world are solid. A solid object consists of solid elements and the process of adding, merging, uniting, or deleting solid elements in a solid object is commonly