Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Bott periodicity, Bott periodicity flavors, Lie group embeddings, Clifford algebras, Complex structures, Octonions Understand how Bott periodicity models the eight divisions of everything Mental Perspectives as Linear Complex Structures. Bott Periodicity Update. Bott Periodicity Models Divisions of Everything Model various features of divisions
Make sense of various aspects of Bott periodicity
Divisions of everything
Lie group inverses
Six transformations
Adjunctions and Conceptions
I am summarizing my investigation of how Bott periodicity models the divisions of everything. The most promising approach is to consider the embeddings of Lie groups. A perspective is modeled by a linear complex structure {$J_k$}. These act on an orthogonal group {$O(16r)$} where {$r$} is an integer. {$J_k^2=1$}. Distinct perspectives {$J_j$} and {$J_k$} are mutually anticommuting linear complex structures, which is to say, {$J_jJ_k=J_kJ_j$}. Thus a set of {$k$} distinct perspectives is understood as the generators of the Clifford algebra {$Cl_{0,k}$}. I am trying to identify a set of {$k$} linear complex structures {$J_1, J_2,\dots ,J_k$} with the division of everything into {$k$} perspectives. As we apply these structures sequentially to {$O(16r)$}, we can ask, which matrices they commute with, and which do they not commute with? In general, it seems that they commute with those which have no reflection, and they do not commute with those which have a reflection. But this notion changes as we keep applying a new, mutually anticommuting structure. Mental reflection is modeled variously, as inversion, as reflection, as complex conjugation, as antilinearity, as that which is culled away because of noncommutativity. Three Minds {$J_\alpha J_\beta$} relates Unconscious and Conscious. {$J_\alpha J_\gamma J_\delta$} divides the vector space into worlds where {$J_\alpha$} and {$J_\beta$} are experienced as How {$v\rightarrow v$} and What {$v\rightarrow v$} (as in the How and What of an arrow by the Yoneda embedding). Whereas {$J_\gamma J_\delta$} is their framework which is not experienced but grounds the experience as Why and Whether. The homotopy groups {$\mathbb{Z}_2$} link the second mind with the first mind, and the third mind with the second mind. The fourth mind is linked to itself, yielding {$\mathbb{Z}$}. So is the eighth mind. Consider Clifford algebras {$Cl_{0,k}$} generated by {$J_1,J_2,\dots ,J_k$}. {$Cl_{0,0}=\mathbb{R}$} Nullsome. {$O(16)$} to {$O(16)$} {$Cl_{0,1}=\mathbb{C}$} Onesome. {$U(8)$} to {$O(16)$} Any linear complex structure functions as a perspective. {$Cl_{0,2}=\mathbb{H}$} Twosome. {$Sp(4)$} to {$O(16)$} Two mutually anticommuting linear complex structures define a quaternionic structure. A product of two linear complex structures functions as a shift in perspective. {$Cl_{0,3}=\mathbb{H}\oplus\mathbb{H}$} Threesome. {$Sp(2)\times Sp(2)$} to {$O(16)$} Pseudoscalar squares to {$+1$}. {$(J_1J_2J_3)^2=1$} We have that {$J_1J_2J_3=J_2J_3J_1=J_2J_3J_1$}. We also have that {$J_1J_2J_3$} splits {$V=V_+\oplus V_$} where {$J_1J_2J_3v_+ = v_+$} for {$v_+\in V_+$} and {$J_1J_2J_3v_ = v_$} for {$v_\in V_$}. {$J_1J_2v_ = J_3v_,\; J_2J_3v_=J_1v_,\; J_3J_1v_=J_2v_$} A product of three linear complex structures breaks up the vector space into two eigenspaces, {$S_+$} and {$S_$}. Three mutually anticommuting linear complex structures define a splitbiquaternionic structure. {$Cl_{0,4}=M_2(\mathbb{H})$} Foursome. {$Sp(2)$} to {$O(16)$} {$J_3J_4$} sets up an isometry between {$V_+$} and {$V_$}. {$Cl_{0,5}=M_4(\mathbb{C})$} Fivesome. {$U(2)$} to {$O(16)$} {$J_1J_4J_5$} commutes with {$J_1J_2J_3$} and acts within {$V_+$} (or {$V_$}) and divides it into mutually orthogonal eigenspaces {$W_\pm$} with {$W_=J_2W_+$}. {$J_2$} interchanges the {$1$} and {$j$} so we are left with those matrices that only mix {$1$} and {$i$}, thus are complex. Can we think of the new operators as developing the gap in between? Or can we think of them as appearing on either end of the chain? {$Cl_{0,6}=M_8(\mathbb{R})$} Sixsome. {$O(2)$} to {$O(16)$} {$J_2J_4J_6$} commutes with {$J_1J_2J_3$} and with {$J_1J_4J_5$} and acts within {$W_\pm$} and splits it into {$X_\pm$} such that {$X_=J_1X_+$}. {$J_1$} acts as {$i$}, thus we are left with those matrices that are without {$i$}. {$Cl_{0,7}=M_8(\mathbb{R})\oplus M_8(\mathbb{R})$} Sevensome. {$O(1)\times O(1)$} to {$O(16)$} {$J_1J_6J_7$} commutes with {$J_1J_2J_3$}, {$J_1J_4J_5$}, {$J_2J_4J_6$} and splits {$X_+$} into subspaces {$Y_\pm$} {$Cl_{0,8}=M_{16}(\mathbb{R})$} Eightsome. {$O(1)$} to {$O(16)$} {$J_7J_8$} commutes with {$J_1J_2J_3$}, {$J_1J_4J_5$}, {$J_2J_4J_6$} but anticommutes with {$J_1J_6J_7$} and thus sets up an isometry between {$Y_+$} and {$Y_$}. Are there circumstances where this could be considered contradictory? Consider flavors of Bott periodicity where this collapses back to the nullsome. In particular, consider Morita equivalence. Yet Morita equivalence equates all matrix algebras of a division algebra, thus equates too many structures. Modeling physical systems Modeling biological systems Krebs cycle. Bott periodicity. Sulfur as an eightcycle. The origin of life.
Ideas
Bott periodicity provides order to the homotopy groups of spheres and that reflects that we are left with Consciousness when we unplug the Unconscious and likewise unplug the Conscious. So the relation between n and m is the relation between the Unconscious and the Conscious and their content. Bott periodicity expresses the symmetry of mathematics
In Bott Periodicity, the matrix representations are related:
Loop spaces
Complex Bott periodicity, which relates U(n) and U(n+m)/U(n)xU(m), and goes back and forth between them, suggests that real Bott periodicity goes back and forth between O(n) and O(n+m)/O(n)xO(m). In one direction there is one step but in the other direction there are seven steps. In Bott periodicity, {$\mathbb{Z}_2$} may count the unconscious and the conscious. They are the two pieces of {$O(\infty)$}. The homotopy groups explain the different ways of relating them. A possible interpretation of the sequence. {$\mathbb{Z}$} expresses a hole as with the threesome and the sevensome. {$\mathbb{Z}_2$} describes the nullsome and the onesome.
