Editing Sedenion (section)

== Arithmetic ==
[[File:Sedenion-Fano Tesseract.gif|thumb|A visualization of a 4D extension to the cubic [[Octonion#Fano plane mnemonic|octonion]],<ref>{{Harv|Baez|2002|loc=p. 6}}</ref> showing the 35 triads as [[hyperplane]]s through the real <math>(e_0)</math> vertex of the sedenion example given]]

Every sedenion is a [[linear combination]] of the unit sedenions <math>e_0</math>, <math>e_1</math>, <math>e_2</math>, <math>e_3</math>, ..., <math>e_{15}</math>,
which form a [[Basis (linear algebra)|basis]] of the [[vector space]] of sedenions. Every sedenion can be represented in the form

:<math>x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{14} e_{14} + x_{15} e_{15}.</math>

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is [[distributive property|distributive]] over addition.

Like other algebras based on the [[Cayley–Dickson construction]], the sedenions contain the algebra they were constructed from. So they contain the octonions (generated by <math>e_0</math> to <math>e_7</math> in the table below), and therefore also the [[quaternion]]s (generated by <math>e_0</math> to <math>e_3</math>), [[complex number]]s (generated by <math>e_0</math> and <math>e_1</math>) and real numbers (generated by <math>e_0</math>).

=== Multiplication ===
Like [[octonion]]s, [[multiplication]] of sedenions is neither [[commutative]] nor [[associative]]. However, in contrast to the octonions, the sedenions do not even have the property of being [[alternative algebra|alternative]]. They do, however, have the property of [[power associativity]], which can be stated as that, for any element <math>x</math> of <math>\mathbb{S}</math>, the power <math>x^n</math> is well defined. They are also [[Flexible algebra|flexible]].

The sedenions have a multiplicative [[identity element]] <math>e_0</math> and multiplicative inverses, but they are not a [[division algebra]] because they have [[zero divisors]]: two nonzero sedenions can be multiplied to obtain zero, for example <math>(e_3 + e_{10})(e_6 - e_{15})</math>. All [[hypercomplex number]] systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

The sedenion multiplication table is shown below:
{| class="wikitable" style="margin:1em auto; text-align: center;"
  !colspan="2" rowspan="2"| <math>e_ie_j</math>
  !colspan="16" |<math>e_j</math>
|-
  ! <math>e_0</math>
  ! <math>e_1</math>
  ! <math>e_2</math>
  ! <math>e_3</math>
  ! <math>e_4</math>
  ! <math>e_5</math>
  ! <math>e_6</math>
  ! <math>e_7</math>
  ! <math>e_8</math>
  ! <math>e_9</math>
  ! <math>e_{10}</math>
  ! <math>e_{11}</math>
  ! <math>e_{12}</math>
  ! <math>e_{13}</math>
  ! <math>e_{14}</math>
  ! <math>e_{15}</math>
|- <!-- Would color coding the cells be nice or annoying? There are too many similar greenish shades though... -->
  ! rowspan="16" | <math>e_i</math>
  ! width="30pt" | <math>e_0</math><!-- color: white; when minus sign -->
  | width="30pt" <!-- style="color: black; background-color: hsla(  0,   0%, 50%, 1);" --> | <math>e_0</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(  0, 100%, 50%, 1);" --> | <math>e_1</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 16, 100%, 50%, 1);" --> | <math>e_2</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 32, 100%, 50%, 1);" --> | <math>e_3</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 48, 100%, 50%, 1);" --> | <math>e_4</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 64, 100%, 50%, 1);" --> | <math>e_5</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 80, 100%, 50%, 1);" --> | <math>e_6</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 96, 100%, 50%, 1);" --> | <math>e_7</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(112, 100%, 50%, 1);" --> | <math>e_8</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(128, 100%, 50%, 1);" --> | <math>e_9</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(144, 100%, 50%, 1);" --> | <math>e_{10}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(160, 100%, 50%, 1);" --> | <math>e_{11}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(176, 100%, 50%, 1);" --> | <math>e_{12}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(192, 100%, 50%, 1);" --> | <math>e_{13}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(208, 100%, 50%, 1);" --> | <math>e_{14}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(224, 100%, 50%, 1);" --> | <math>e_{15}</math>
|-
  ! <math>e_1</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
|-
  ! <math>e_2</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_4</math>
  | <math>-e_5</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
|-
  ! <math>e_3</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
|-
  ! <math>e_4</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
|-
  ! <math>e_5</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
|-
  ! <math>e_6</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_8</math>
  | <math>e_9</math>
|-
  ! <math>e_7</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>-e_8</math>
|-
  ! <math>e_8</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_{12}</math>
  | <math>-e_{13}</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
|-
  ! <math>e_9</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>-e_5</math>
  | <math>e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
|-
  ! <math>e_{10}</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>e_4</math>
  | <math>e_5</math>
|-
  ! <math>e_{11}</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>e_4</math>
|-
  ! <math>e_{12}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_2</math>
  | <math>-e_3</math>
|-
  ! <math>e_{13}</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
|-
  ! <math>e_{14}</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>e_8</math>
  | <math>e_9</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
|-
  ! <math>e_{15}</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>e_8</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
|}

=== Sedenion properties ===
[[File:PG(3,2) g005.png|thumb|right|An illustration of the structure of [[PG(3,2)]] that provides the multiplication law for sedenions, as shown by {{harvtxt|Saniga|Holweck|Pracna|2015}}. Any three points (representing three sedenion imaginary units) lying on the same line are such that the product of two of them yields the third one, sign disregarded.]]

From the above table, we can see that:

:<math>e_0e_i = e_ie_0 = e_i \, \text{for all} \, i,</math>

:<math>e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0,</math> and

:<math>e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{with}\,\, i,j \neq 0.</math>

==== Anti-associative ====
The sedenions are not fully anti-associative. Choose any four generators, <math>i,j,k</math> and <math>l</math>. The following 5-cycle shows that these five relations cannot all be anti-associative.

<math display="block">(ij)(kl) = -((ij)k)l = (i(jk))l = -i((jk)l) = i(j(kl)) = -(ij)(kl) = 0</math>

In particular, in the table above, using <math>e_1,e_2,e_4</math> and <math>e_8</math> the last expression associates.
<math>(e_1e_2)e_{12} = e_1(e_2e_{12}) = -e_{15}</math>

=== Quaternionic subalgebras ===
The particular sedenion multiplication table shown above is represented by 35 triads. The table and its triads have been constructed from an [[octonion]] represented by the bolded set of 7 triads using [[Cayley–Dickson construction]]. It is one of 480 possible sets of 7 triads (one of two shown in the octonion article) and is the one based on the Cayley–Dickson construction of [[quaternions]] from two possible quaternion constructions from the [[complex numbers]]. The binary representations of the indices of these triples [[bitwise XOR]] to 0. These 35 triads are:

{ '''{1, 2, 3}''', '''{1, 4, 5}''', '''{1, 7, 6}''', {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15}, <br />
'''{2, 4, 6}''', '''{2, 5, 7}''', {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, '''{3, 4, 7}''', <br />
'''{3, 6, 5}''', {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},  <br />
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},  <br />
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

=== Zero divisors ===
The list of 84 sets of zero divisors <math>\{e_a, e_b, e_c, e_d\}</math>, where
<math>(e_a + e_b) \circ (e_c + e_d) = 0</math>:

<math display="block">\begin{array}{c}
\text{Sedenion Zero Divisors} \quad \{ e_a, e_b, e_c, e_d \} \\
\text{where} ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\
\begin{array}{ccc}
1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\
\end{array} \\
\\ 
\begin{array}{lccr}
\{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \} &
\{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \}\\
\end{array} \\ 
\\
\begin{array}{lccr}
\{e_1, e_{10}, e_5, e_{14}\} & \{e_1, e_{10}, e_4, -e_{15}\} &
\{e_1, e_{10}, e_7, e_{12}\} & \{e_1, e_{10}, e_6, -e_{13}\} \\
\{e_1, e_{11}, e_4, e_{14}\} & \{e_1, e_{11}, e_6, -e_{12}\} &
\{e_1, e_{11}, e_5, e_{15}\} & \{e_1, e_{11}, e_7, -e_{13}\} \\
\{e_1, e_{12}, e_2, e_{15}\} & \{e_1, e_{12}, e_3, -e_{14}\} &
\{e_1, e_{12}, e_6, e_{11}\} & \{e_1, e_{12}, e_7, -e_{10}\} \\
\{e_1, e_{13}, e_6, e_{10}\} & \{e_1, e_{13}, e_2, -e_{14}\} &
\{e_1, e_{13}, e_7, e_{11}\} & \{e_1, e_{13}, e_3, -e_{15}\} \\
\{e_1, e_{14}, e_2, e_{13}\} & \{e_1, e_{14}, e_4, -e_{11}\} &
\{e_1, e_{14}, e_3, e_{12}\} & \{e_1, e_{14}, e_5, -e_{10}\} \\
\{e_1, e_{15}, e_3, e_{13}\} & \{e_1, e_{15}, e_2, -e_{12}\} &
\{e_1, e_{15}, e_4, e_{10}\} & \{e_1, e_{15}, e_5, -e_{11}\} \\
\\ 
\{e_2, e_9,    e_4, e_{15}\} & \{e_2, e_9,    e_5, -e_{14}\} &
\{e_2, e_9,    e_6, e_{13}\} & \{e_2, e_9,    e_7, -e_{12}\} \\
\{e_2, e_{11}, e_5, e_{12}\} & \{e_2, e_{11}, e_4, -e_{13}\} &
\{e_2, e_{11}, e_6, e_{15}\} & \{e_2, e_{11}, e_7, -e_{14}\} \\
\{e_2, e_{12}, e_3, e_{13}\} & \{e_2, e_{12}, e_5, -e_{11}\} &
\{e_2, e_{12}, e_7, e_9   \} & \{e_2, e_{13}, e_3, -e_{12}\} \\
\{e_2, e_{13}, e_4, e_{11}\} & \{e_2, e_{13}, e_6, -e_9   \} &
\{e_2, e_{14}, e_5, e_9   \} & \{e_2, e_{14}, e_3, -e_{15}\} \\
\{e_2, e_{14}, e_7, e_{11}\} & \{e_2, e_{15}, e_4, -e_9   \} &
\{e_2, e_{15}, e_3, e_{14}\} & \{e_2, e_{15}, e_6, -e_{11}\} \\
\\ 
\{e_3, e_9,    e_6, e_{12}\} & \{e_3, e_9,    e_4, -e_{14}\} &
\{e_3, e_9,    e_7, e_{13}\} & \{e_3, e_9,    e_5, -e_{15}\} \\
\{e_3, e_{10}, e_4, e_{13}\} & \{e_3, e_{10}, e_5, -e_{12}\} &
\{e_3, e_{10}, e_7, e_{14}\} & \{e_3, e_{10}, e_6, -e_{15}\} \\
\{e_3, e_{12}, e_5, e_{10}\} & \{e_3, e_{12}, e_6, -e_9   \} &
\{e_3, e_{14}, e_4, e_9   \} & \{e_3, e_{13}, e_4, -e_{10}\} \\
\{e_3, e_{15}, e_5, e_9   \} & \{e_3, e_{13}, e_7, -e_9   \} &
\{e_3, e_{15}, e_6, e_{10}\} & \{e_3, e_{14}, e_7, -e_{10}\} \\
\\ 
\{e_4, e_9,    e_7, e_{10}\} & \{e_4, e_9,    e_6, -e_{11}\} &
\{e_4, e_{10}, e_5, e_{11}\} & \{e_4, e_{10}, e_7, -e_9   \} \\
\{e_4, e_{11}, e_6, e_9   \} & \{e_4, e_{11}, e_5, -e_{10}\} &
\{e_4, e_{13}, e_6, e_{15}\} & \{e_4, e_{13}, e_7, -e_{14}\} \\
\{e_4, e_{14}, e_7, e_{13}\} & \{e_4, e_{14}, e_5, -e_{15}\} &
\{e_4, e_{15}, e_5, e_{14}\} & \{e_4, e_{15}, e_6, -e_{13}\} \\
\\ 
\{e_5, e_{10}, e_6, e_9   \} & \{e_5, e_9,    e_6, -e_{10}\} &
\{e_5, e_{11}, e_7, e_9   \} & \{e_5, e_9,    e_7, -e_{11}\} \\
\{e_5, e_{12}, e_7, e_{14}\} & \{e_5, e_{12}, e_6, -e_{15}\} &
\{e_5, e_{15}, e_6, e_{12}\} & \{e_5, e_{14}, e_7, -e_{12}\} \\
\\ 
\{e_6, e_{11}, e_7, e_{10}\} & \{e_6, e_{10}, e_7, -e_{11}\} &
\{e_6, e_{13}, e_7, e_{12}\} & \{e_6, e_{12}, e_7, -e_{13}\}
\end{array}
\end{array}</math>