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For 5 dimensions can be generated by 5 basis vectors, e1, e2, e3, e4 and e5
One of the most important applications of a Geometric Algebra based on 5D vector space is to represent conformal space.
In the case of conformal space e1, e2, e3, e0 and e∞ so e4 and e5 can be replaced by e0 and e∞ where:
:
| grade | base value | numerical value | |
|---|---|---|---|
full |
shortened |
||
| 0=unit scalar | 1 | e | |
| 1=unit length base vectors | e1 | e1 | |
| e2 | e2 | ||
| e3 | e3 | ||
| e4 | e4 | ||
| e5 | e5 | ||
| 2=unit length base bivectors | e1^ e2 | e12 | e12 |
| e3^ e1 | e31 | e31 | |
| e2^ e3 | e23 | e23 | |
| e1^ e4 | e14 | e14 | |
| e4^ e2 | e42 | e42 | |
| e3^ e4 | e34 | e34 | |
| e1^ e5 | e15 | e15 | |
| e2^ e5 | e25 | e25 | |
| e3^ e5 | e35 | e35 | |
| e4^ e5 | e45 | e45 | |
| 3=unit length base tri-vector | e1^ e2^ e3 | e123 | e123 |
| e2^ e1^ e4 | e214 | e214 | |
| e1^ e4^ e3 | e143 | e143 | |
| e2^ e3^ e4 | e234 | e234 | |
| e1^ e2^ e5 | e125 | e125 | |
| e1^ e3^ e5 | e135 | e135 | |
| e2^ e3^ e5 | e235 | e235 | |
| e1^ e4^ e5 | e145 | e145 | |
| e2^ e4^ e5 | e245 | e245 | |
| e3^ e4^ e5 | e345 | e345 | |
| 4=unit length base quad-vector | e1^ e2^ e3^ e4 | e1234 | e1234 |
| e1^ e2^ e3^ e5 | e1235 | e1235 | |
| e1^ e2^ e4^ e5 | e1245 | e1245 | |
| e1^ e3^ e4^ e5 | e1345 | e1345 | |
| e2^ e3^ e4^ e5 | e2345 | e2345 | |
| 5=unit length base pent-vector | e1^ e2^ e3^ e4^ e5 | e12345 | e12345 |
So in this case the number of dimensions is:
In this case the number of scalar values in the multivector is 32 = (1+5+10+10+5+1)
You may be interested in other means to represent orientation and rotational quantities such as:
Or you may be interested in how these quantities are used to simulate physical objects:
If you are not familiar with vectors this subject you may like to look at the following page first:
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Could anyone let me know of a good proof that a quaternion multiplication can be used to represent a rotation in 3 dimensions, I'm not looking for the shortest proof, but the most easily understood. |
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