Maths - bases - 4D

Here is how I generated the tables for this page.

The tables were generated using this program.

The output of this program is shown below. To produce the results the program needs to have an XML input code. At the bottom of this page I have listed this input code.

Table for: g 4,0,0

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1ee12-e31-e41e2-e3e123-e4e142-e134e23e42-e43-e1234-e324
a.e2e2-e12ee23-e42-e1e123e3-e142-e4e324e31-e41-e1234e43-e134
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e12-e2e1e123-e142-ee23-e31-e42e41-e1234-e3e4e324-e134e43
a.e31e31e3e123-e1-e134-e23-ee12e43-e1234-e41-e2-e324e4e142e42
a.e23e23e123-e3e2-e324e31-e12-e-e1234-e43e42-e1e134-e142e4e41
a.e41e41e4-e142e134-e1e42-e43-e1234-e-e12e31-e324e2-e3e123e23
a.e42e42e142e4-e324-e2-e41-e1234e43e12-e-e23-e134-e1e123e3e31
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e123e23e31e12e1234-e3-e2-e1e324e134e142-e-e43-e42-e41-e4
a.e142e142e42-e41e1234-e12e4e324-e134e2-e1-e123e43-ee23-e31-e3
a.e134e134-e43e1234e41-e31-e324e4e142-e3-e123e1e42-e23-ee12-e2
a.e324e324e1234e43-e42-e23e134-e142e4-e123e3-e2e41e31-e12-e-e1
a.e1234e1234e324e134e142e123e43e42e41e23e31e12e4e3e2e1e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 4,0,0

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e12-ee23-e31-e42e41-e1234e43
a.e31e31-e23-ee12e43-e1234-e41e42
a.e23e23e31-e12-e-e1234-e43e42e41
a.e41e41e42-e43-e1234-e-e12e31e23
a.e42e42-e41-e1234e43e12-e-e23e31
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e1234e43e42e41e23e31e12e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 3,1,0

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1-ee12-e31-e41-e2e3e123e4e142-e134-e23-e42e43-e1234e324
a.e2e2-e12ee23-e42-e1e123e3-e142-e4e324e31-e41-e1234e43-e134
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e12e2e1e123-e142e-e23-e31e42e41-e1234e3-e4-e324-e134-e43
a.e31e31-e3e123-e1-e134e23ee12-e43-e1234-e41e2e324-e4e142-e42
a.e23e23e123-e3e2-e324e31-e12-e-e1234-e43e42-e1e134-e142e4e41
a.e41e41-e4-e142e134-e1-e42e43-e1234e-e12e31e324-e2e3e123-e23
a.e42e42e142e4-e324-e2-e41-e1234e43e12-e-e23-e134-e1e123e3e31
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e123-e23e31e12e1234e3e2-e1-e324e134e142ee43e42-e41e4
a.e142e142-e42-e41e1234-e12-e4-e324-e134-e2-e1-e123-e43e-e23-e31e3
a.e134e134e43e1234e41-e31e324-e4e142e3-e123e1-e42e23ee12e2
a.e324e324e1234e43-e42-e23e134-e142e4-e123e3-e2e41e31-e12-e-e1
a.e1234e1234-e324e134e142e123-e43-e42e41-e23e31e12-e4-e3-e2e1-e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 3,1,0

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e12e-e23-e31e42e41-e1234-e43
a.e31e31e23ee12-e43-e1234-e41-e42
a.e23e23e31-e12-e-e1234-e43e42e41
a.e41e41-e42e43-e1234e-e12e31-e23
a.e42e42-e41-e1234e43e12-e-e23e31
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e1234-e43-e42e41-e23e31e12-e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 2,2,0

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1-ee12-e31-e41-e2e3e123e4e142-e134-e23-e42e43-e1234e324
a.e2e2-e12-ee23-e42e1e123-e3-e142e4e324-e31e41-e1234-e43e134
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e12e2-e1e123-e142-e-e23e31e42-e41-e1234-e3e4-e324e134e43
a.e31e31-e3e123-e1-e134e23ee12-e43-e1234-e41e2e324-e4e142-e42
a.e23e23e123e3e2-e324-e31-e12e-e1234e43e42e1-e134-e142-e4-e41
a.e41e41-e4-e142e134-e1-e42e43-e1234e-e12e31e324-e2e3e123-e23
a.e42e42e142-e4-e324-e2e41-e1234-e43e12e-e23e134e1e123-e3-e31
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e123-e23-e31e12e1234-e3e2e1-e324-e134e142-e-e43e42e41-e4
a.e142e142-e42e41e1234-e12e4-e324e134-e2e1-e123e43-e-e23e31-e3
a.e134e134e43e1234e41-e31e324-e4e142e3-e123e1-e42e23ee12e2
a.e324e324e1234-e43-e42-e23-e134-e142-e4-e123-e3-e2-e41-e31-e12ee1
a.e1234e1234-e324-e134e142e123e43-e42-e41-e23-e31e12e4e3-e2-e1e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 2,2,0

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e12-e-e23e31e42-e41-e1234e43
a.e31e31e23ee12-e43-e1234-e41-e42
a.e23e23-e31-e12e-e1234e43e42-e41
a.e41e41-e42e43-e1234e-e12e31-e23
a.e42e42e41-e1234-e43e12e-e23-e31
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e1234e43-e42-e41-e23-e31e12e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 0,4,0

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1-ee12-e31-e41-e2e3e123e4e142-e134-e23-e42e43-e1234e324
a.e2e2-e12-ee23-e42e1e123-e3-e142e4e324-e31e41-e1234-e43e134
a.e3e3e31-e23-e-e43e123-e1e2e134-e324e4-e12-e1234-e41e42e142
a.e4e4e41e42e43-e-e142-e134-e324-e1-e2-e3-e1234e12e31e23e123
a.e12e12e2-e1e123-e142-e-e23e31e42-e41-e1234-e3e4-e324e134e43
a.e31e31-e3e123e1-e134e23-e-e12-e43-e1234e41-e2e324e4-e142e42
a.e23e23e123e3-e2-e324-e31e12-e-e1234e43-e42-e1-e134e142e4e41
a.e41e41-e4-e142e134e1-e42e43-e1234-ee12-e31e324e2-e3-e123e23
a.e42e42e142-e4-e324e2e41-e1234-e43-e12-ee23e134-e1-e123e3e31
a.e43e43-e134e324-e4e3-e1234-e41e42e31-e23-ee142-e123e1-e2e12
a.e123e123-e23-e31-e12e1234-e3-e2-e1-e324-e134-e142e-e43-e42-e41e4
a.e142e142-e42e41e1234e12e4-e324e134e2-e1e123e43ee23-e31e3
a.e134e134e43e1234-e41e31e324e4-e142-e3e123e1e42-e23ee12e2
a.e324e324e1234-e43e42e23-e134e142e4e123e3-e2e41e31-e12ee1
a.e1234e1234-e324-e134-e142-e123e43e42e41e23e31e12-e4-e3-e2-e1e

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 0,4,0

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e12-e-e23e31e42-e41-e1234e43
a.e31e31e23-e-e12-e43-e1234e41e42
a.e23e23-e31e12-e-e1234e43-e42e41
a.e41e41-e42e43-e1234-ee12-e31e23
a.e42e42e41-e1234-e43-e12-ee23e31
a.e43e43-e1234-e41e42e31-e23-ee12
a.e1234e1234e43e42e41e23e31e12e

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 3,0,1

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1ee12-e31-e41e2-e3e123-e4e142-e134e23e42-e43-e1234-e324
a.e2e2-e120e23-e420e1230-e1420e32400-e123400
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e12-e20e123-e1420e230-e420-e123400e32400
a.e31e31e3e123-e1-e134-e23-ee12e43-e1234-e41-e2-e324e4e142e42
a.e23e23e1230e2-e3240-e120-e12340e4200-e14200
a.e41e41e4-e142e134-e1e42-e43-e1234-e-e12e31-e324e2-e3e123e23
a.e42e42e1420-e324-e20-e12340e120-e2300e12300
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e123e230e12e12340-e20e3240e14200-e4200
a.e142e142e420e1234-e120e3240e20-e12300e2300
a.e134e134-e43e1234e41-e31-e324e4e142-e3-e123e1e42-e23-ee12-e2
a.e324e324e12340-e42-e230-e1420-e1230-e200-e1200
a.e1234e1234e3240e142e1230e420e230e1200e200

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 3,0,1

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e120e230-e420-e12340
a.e31e31-e23-ee12e43-e1234-e41e42
a.e23e230-e120-e12340e420
a.e41e41e42-e43-e1234-e-e12e31e23
a.e42e420-e12340e120-e230
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e12340e420e230e120

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 2,1,1

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e1-ee12-e31-e41-e2e3e123e4e142-e134-e23-e42e43-e1234e324
a.e2e2-e120e23-e420e1230-e1420e32400-e123400
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e12e20e123-e1420-e230e420-e123400-e32400
a.e31e31-e3e123-e1-e134e23ee12-e43-e1234-e41e2e324-e4e142-e42
a.e23e23e1230e2-e3240-e120-e12340e4200-e14200
a.e41e41-e4-e142e134-e1-e42e43-e1234e-e12e31e324-e2e3e123-e23
a.e42e42e1420-e324-e20-e12340e120-e2300e12300
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e123-e230e12e12340e20-e3240e14200e4200
a.e142e142-e420e1234-e120-e3240-e20-e12300-e2300
a.e134e134e43e1234e41-e31e324-e4e142e3-e123e1-e42e23ee12e2
a.e324e324e12340-e42-e230-e1420-e1230-e200-e1200
a.e1234e1234-e3240e142e1230-e420-e230e1200-e200

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 2,1,1

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e120-e230e420-e12340
a.e31e31e23ee12-e43-e1234-e41-e42
a.e23e230-e120-e12340e420
a.e41e41-e42e43-e1234e-e12e31-e23
a.e42e420-e12340e120-e230
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e12340-e420-e230e120

analysing commutivity: table does not commute: for example: e12*e31 != e31*e12

analysing associativity: table associates

Table for: g 2,0,2

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e10e12-e31-e4100e1230e142-e134000-e12340
a.e2e2-e120e23-e420e1230-e1420e32400-e123400
a.e3e3e31-e23e-e43e123e1-e2e134-e324-e4e12-e1234e41-e42-e142
a.e4e4e41e42e43e-e142-e134-e324e1e2e3-e1234-e12-e31-e23-e123
a.e12e1200e123-e14200000-e123400000
a.e31e310e123-e1-e13400e120-e1234-e41000e1420
a.e23e23e1230e2-e3240-e120-e12340e4200-e14200
a.e41e410-e142e134-e100-e12340-e12e31000e1230
a.e42e42e1420-e324-e20-e12340e120-e2300e12300
a.e43e43-e134e324e4-e3-e1234e41-e42-e31e23-e-e142e123e1-e2e12
a.e123e12300e12e123400000e14200000
a.e142e14200e1234-e1200000-e12300000
a.e134e1340e1234e41-e3100e1420-e123e1000e120
a.e324e324e12340-e42-e230-e1420-e1230-e200-e1200
a.e1234e123400e142e12300000e1200000

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

Table for: g+ 2,0,2

a*bb.eb.e12b.e31b.e23b.e41b.e42b.e43b.e1234
a.eee12e31e23e41e42e43e1234
a.e12e1200000-e12340
a.e31e3100e120-e1234-e410
a.e23e230-e120-e12340e420
a.e41e4100-e12340-e12e310
a.e42e420-e12340e120-e230
a.e43e43-e1234e41-e42-e31e23-ee12
a.e1234e123400000e120

analysing commutivity: table does not commute: for example: e31*e23 != e23*e31

analysing associativity: table associates

Table for: g 0,0,4

a*bb.eb.e1b.e2b.e3b.e4b.e12b.e31b.e23b.e41b.e42b.e43b.e123b.e142b.e134b.e324b.e1234
a.eee1e2e3e4e12e31e23e41e42e43e123e142e134e324e1234
a.e1e10e12-e31-e4100e1230e142-e134000-e12340
a.e2e2-e120e23-e420e1230-e1420e32400-e123400
a.e3e3e31-e230-e43e12300e134-e32400-e1234000
a.e4e4e41e42e430-e142-e134-e324000-e12340000
a.e12e1200e123-e14200000-e123400000
a.e31e310e1230-e1340000-e1234000000
a.e23e23e12300-e324000-e12340000000
a.e41e410-e142e134000-e123400000000
a.e42e42e1420-e32400-e1234000000000
a.e43e43-e134e32400-e12340000000000
a.e123e123000e123400000000000
a.e142e14200e1234000000000000
a.e134e1340e12340000000000000
a.e324e324e123400000000000000
a.e1234e1234000000000000000

analysing commutivity: table does not commute: for example: e1*e2 != e2*e1

analysing associativity: table associates

XML input code

To produce the results the program needs to have an XML input code listed here:

<classDef>
<outputTable type="product" format="html" name="g 4,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="4" sign="0" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 4,0,0" analyse="on" enableLabels="on">
<mathTypeMulti name="a" type="4" sign="0" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 3,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 3,1,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="1" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 2,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="3" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 2,2,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="3" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 0,4,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="15" zero="0" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 0,4,0" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="15" zero="0" subAlgebra="even"/>
</outputTable>
<outputTable type="product" format="html" name="g 3,0,1" analyse="on" enableLabels="on">
<mathTypeMulti name="b" type="4" sign="0" zero="2" subAlgebra="all"/>
</outputTable>
<outputTable type="product" format="html" name="g+ 3,0,1" analyse="on" enableLabels="on">
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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

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