--- title: "Definition of Kime" subtitle: "[Back To TCIU Contents](https://tciu.predictive.space/)" author: "SOCR Team " date: "`r format(Sys.time(),'%m/%d/%Y')`" output: html_document: theme: spacelab highlight: tango includes: before_body: TCIU_header.html toc: true number_sections: true toc_depth: 2 toc_float: collapsed: false smooth_scroll: true code_folding: hide --- ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE, warings = FALSE) ``` # Figure 1.8 A schematic of the 5D spacekime manifold, two cones representing space ($\mathbb{R}^3$) - kime ($\mathbb{R}^2$) Complex-time (kime) can be defined using [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinate_system), $\kappa = t\ e^{i \varphi}$. ```{r message=F, warning=F} library(plotly) library(dplyr) ``` *Kime* naturally extends *time*, $\kappa = t\ e^{i \varphi}$, i.e., time is just the *magnitude* of kime, $t=|\kappa|$. The time domain $(t\in\mathbb{R}^+)$ is a subgroup of the multiplicative group of the reals, whereas kime $(\kappa\in\mathbb{C})$ is the smallest algebraically closed prime field that naturally extends the time domain. The time domain of the positive reals is *ordered*, but its kime extension is *not.* The kime domain, $\mathbb{C}$, represents the smallest natural extension of time, as a complete filed that covers time. This Figure shows the spacekime representation of observable processes indexed by space, reduced to 1D, and kime (2D). At a given spatial location ($x$), the red balls scattered along the circle of fixed radius (time) represent the repeated process measurements from the experimentally controlled trials colocalized in spacetime. ```{r fig1.8, message=F, warning=F} # parameter space sweep for the spherical coordinates phi <- seq(from = 0, to = 2*pi, by = ((2*pi - 0)/(200 - 1))) psi <- seq(from = 0, to = pi, by = ((pi - 0)/(200 - 1))) # shape=="cone1") # rendering (u,v) parametric surfaces requires x,y,z arguments to be 2D arrays # In out case, the three coordinates have to be 200*200 parameterized tensors/arrays h1= 10 # cone height r1 = seq(from = 0, to = h1, by = ((h1 - 0)/(200 - 1))) # r = radius x1 = 20* ((h1 - r1)/h1 ) %o% rep(1, 200) # x = 3*r y1 = 3* ((h1 - r1)/h1 ) %o% sin(phi) # y = r*sin(phi) z1 = 3* ((h1 - r1)/h1 ) %o% cos(phi) # z = r*cos(phi) # circle1 boundary x11 = rep(20, 200) %o% rep(1, 200) # x = 20 y11 = 3* ((h1 - r1)/h1 ) %o% sin(phi) # y = r*sin(phi) z11 = 3* ((h1 - r1)/h1 ) %o% cos(phi) # z = r*cos(phi) # randomly kime-phase sample points on cone3 randX1 <- sample((dim(x1))[2], 10) # shape=="cone2") h2= 10 # cone height r2 = seq(from = 0, to = h2, by = ((h2 - 0)/(200 - 1))) # r = radius x2 = 20* ((h2 - r2)/h2 ) %o% rep(1, 200) # x = 3*r y2 = 2* ((h2 - r2)/h2 ) %o% sin(phi) # y = r*sin(phi) z2 = 2* ((h2 - r2)/h2 ) %o% cos(phi) # z = r*cos(phi) # circle2 boundary x21 = rep(20, 200) %o% rep(1, 200) # x = 20 y21 = 2* ((h2 - r2)/h2 ) %o% sin(phi) # y = r*sin(phi) z21 = 2* ((h2 - r2)/h2 ) %o% cos(phi) # z = r*cos(phi) # shape=="cone3") h3= 10 # cone height r3 = seq(from = 0, to = h3, by = ((h3 - 0)/(200 - 1))) # r = radius x3 = 15* ((h3 - r3)/h3 ) %o% rep(1, 200) # x = 3*r y3 = 3* ((h3 - r3)/h3 ) %o% sin(phi) # y = r*sin(phi) z3 = 3* ((h3 - r3)/h3 ) %o% cos(phi) # z = r*cos(phi) # circle3 boundary x31 = rep(15, 200) %o% rep(1, 200) # x = 15 y31 = 3* ((h3 - r3)/h3) %o% sin(phi) # y = r*sin(phi) z31 = 3* ((h3 - r3)/h3) %o% cos(phi) # z = r*cos(phi) shape_names <- c("all", "cone1", "cone2", "cone3") # https://plot.ly/r/custom-buttons/ # updatemenus component updatemenus <- list( list( active = -1, type = 'buttons', buttons = list( list( label = shape_names[1], method = "update", args = list(list(visible = c(TRUE, TRUE, TRUE, TRUE)), list(title = shape_names[1]))), list( label = shape_names[2], method = "update", args = list(list(visible = c(TRUE, FALSE, FALSE, FALSE)), list(title = shape_names[2]))), list( label = shape_names[3], method = "update", args = list(list(visible = c(TRUE, TRUE, FALSE, FALSE)), list(title = shape_names[3]))), list( label = shape_names[4], method = "update", args = list(list(visible = c(TRUE, FALSE, FALSE, TRUE)), list(title = shape_names[4]))) ) ) ) p <- plot_ly(showscale = FALSE) %>% # add cone1 # Randomly sample points (kime-phase sampling) the boundary of cone1 surface add_trace(x=~x1[1,randX1], y=~y11[1,randX1], z=~z11[1,randX1], type="scatter3d", mode="markers", marker = list(size = 10, color="red"), name="Phase-Samples", text = paste0("Kime-Phase:\n", " space=", 20, "\n time=|kime|=", h1, "\n kime-phase=", round((phi[randX1])-pi,2))) %>% add_trace(x = ~x1, y = ~y1, z = ~z1, type = 'surface', opacity=0.3, visible=T, contour=list(show=F, color="#000", width=15, lwd=10, opacity=0.5, hoverinfo="none", legendshow=F)) %>% # add cone2 add_trace(x = ~x2, y = ~y2, z = ~z2, type='surface', opacity=0.4,visible=T, contour=list(show=F, color="#000", width=15, lwd=10, opacity=0.5, hoverinfo="none", legendshow=F)) %>% # add cone3 add_trace(x = ~x3, y = ~y3, z = ~z3, type='surface', opacity=0.5,visible=T, contour=list(show=F, color="#000", width=15, lwd=10, opacity=0.5, hoverinfo="none", legendshow=F)) %>% # # # trace the x-axis add_trace(x=~1.1*x1[,1], y=0, z=0, type="scatter3d", mode="lines", line = list(width = 10, color="light blue"), name="Z", hoverinfo="none", legendshow=F) %>% # # # trace the boundary of cone1 surface add_trace(x=~x1[1,], y=~y11[1,], z=~z11[1,], type="scatter3d", mode="lines", line = list(width = 10, color="red"), name="Surface Boundary", hoverinfo="none", legendshow=F) %>% # add center for kime circle1 at location x1 add_trace(x=~x1[1,1], y=0, z=0, type="scatter3d", mode="markers", marker = list(size = 10, color="red"), name="Z", hoverinfo="none", legendshow=F) %>% # trace the boundary of cone2 surface add_trace(x=~x2[1,], y=~y21[1,], z=~z21[1,], type="scatter3d", mode="lines", line = list(width = 10, color="green"), name="Surface Boundary", hoverinfo="none", legendshow=F) %>% # trace the boundary of cone3 surface add_trace(x=~x3[1,], y=~y31[1,], z=~z31[1,], type="scatter3d", mode="lines", line = list(width = 10, color="blue"), name="Surface Boundary", hoverinfo="none", legendshow=F) %>% # add center for cime circle3 at location x3 add_trace(x=~x3[1,1], y=0, z=0, type="scatter3d", mode="markers", line = list(width = 10, color="navy blue"), name="Z", hoverinfo="none", legendshow=F) %>% # layout layout(title = "Schematic of Space (1D) and Kime (2D) Representaiton", showlegend = FALSE, scene = list(xaxis=list(title="space"), yaxis=list(title="kappa1"), zaxis=list(title="kappa2")), updatemenus = updatemenus) p ``` # Kime-Phase Representation See [this kime-phase representation GIF animation](https://socr.umich.edu/docs/uploads/2025/SOCR_KimePhase_Animation_2025.gif). ```{r KimePhaseRep, message=F, warning=F} library(animation) library(plotly) library(circular) epsilon <- 0.1 sampleSize <- 1000 # total number of phases to sample for 3 different processes (x, y, z) sizePerTime <- 100 # number of phases to use for each fixed time (must divide sampleSize) circleUniformPhi <- seq(from=-pi, to=pi, length.out=sizePerTime) oopt = ani.options(interval = 0.2) set.seed(1234) # sample the the kime-phases for all 3 different processes and the r time points x <- rvonmises(n=sampleSize, mu=circular(pi/5), kappa=3) y <- rvonmises(n=sampleSize, mu=circular(-pi/3), kappa=5) z <- rvonmises(n=sampleSize, mu=circular(0), kappa=10) r <- seq(from=1, to=sampleSize/sizePerTime, length.out=10) # Define a function that renormalizes the kime-phase to [-pi, pi) pheRenormalize <- function (x) { out <- ifelse(as.numeric(x) <= pi, as.numeric(x)+pi, as.numeric(x)-pi) return (out) } # transform Von Mises samples from [0, 2*pi) to [-pi, pi) x <- pheRenormalize(x) y <- pheRenormalize(y) z <- pheRenormalize(z) # vectorize the samples vectorX = as.vector(x) vectorY = as.vector(y) vectorZ = as.vector(z) # Starting phases, set the first phase index=1 plotX = c(vectorX[1]) plotY = c(vectorY[1]) plotZ = c(vectorZ[1]) pl_scene <- plot_ly(type='scatter3d', mode="markers") plotX <- list() plotY <- list() plotZ <- list() plotX_df <- list() # need separate dataframes to store all time foliations plotY_df <- list() plotZ_df <- list() for (t in 1:length(r)) { # loop over time # loop over kime-phases plotX[[t]] <- as.numeric(x[c(( (t-1)*length(r) + 1):((t-1)*length(r) + sizePerTime))]) plotY[[t]] <- as.numeric(y[c(( (t-1)*length(r) + 1):((t-1)*length(r) + sizePerTime))]) plotZ[[t]] <- as.numeric(z[c(( (t-1)*length(r) + 1):((t-1)*length(r) + sizePerTime))]) # Try to "stack=T the points .... #r1 = sqrt((resx$x)^2+(resx$y)^2)/2; #resx$x = r1*cos(resx$data) #resx$x = r1*cos(resx$data) tempX = circular(unlist(plotX[[t]])) tempY = circular(unlist(plotY[[t]])) tempZ = circular(unlist(plotZ[[t]])) resx <- density(tempX, bw=25, xaxt='n', yaxt='n') resy <- density(tempY, bw=25, xaxt='n', yaxt='n') resz <- density(tempZ, bw=25, xaxt='n', yaxt='n') # res <- plot(resx, points.plot=TRUE, xlim=c(-1.1,1.5), ylim=c(-1.5, 1.5), # main="Trivariate random sampling of\n kime-magnitudes (times) and kime-directions (phases)", # offset=0.9, shrink=1.0, ticks=T, lwd=3, axes=F, stack=TRUE, bins=150) # pl_list[[t]] unifPhi_df <- as.data.frame(cbind(t=t, circleUniformPhi=circleUniformPhi)) plotX_df[[t]] <- as.data.frame(cbind(t=t, plotX=unlist(plotX[[t]]))) plotY_df[[t]] <- as.data.frame(cbind(t=t, plotY=unlist(plotY[[t]]))) plotZ_df[[t]] <- as.data.frame(cbind(t=t, plotZ=unlist(plotZ[[t]]))) pl_scene <- pl_scene %>% add_trace(data=unifPhi_df, showlegend=FALSE, x = ~((t-epsilon)*cos(circleUniformPhi)), y = ~((t-epsilon)*sin(circleUniformPhi)), z=0, name=paste0("Time=",t), line=list(color='gray'), mode = 'lines', opacity=0.3) %>% add_markers(data=plotX_df[[t]], x=~(t*cos(plotX)), y=~(t*sin(plotX)), z=0, type='scatter3d', name=paste0("X: t=",t), marker=list(color='green'), showlegend=FALSE, mode = 'markers', opacity=0.3) %>% add_markers(data=plotY_df[[t]], x=~((t+epsilon)*cos(plotY)), y=~((t+epsilon)*sin(plotY)), z=0-epsilon, showlegend=FALSE, type='scatter3d', name=paste0("Y: t=",t), marker=list(color='blue'), mode = 'markers', opacity=0.3) %>% add_markers(data=plotZ_df[[t]], x=~((t+2*epsilon)*cos(plotZ)), y=~((t+2*epsilon)*sin(plotZ)), z=0+epsilon, showlegend=FALSE, type='scatter3d', name=paste0("Z: t=",t), marker=list(color='red'), mode = 'markers', opacity=0.3) } means_df <- as.data.frame(cbind(t = c(1:length(r)), plotX_means=unlist(lapply(plotX, mean)), plotY_means=unlist(lapply(plotY, mean)), plotZ_means=unlist(lapply(plotZ, mean)))) pl_scene <- pl_scene %>% # add averaged (donoised) phase trajectories add_trace(data=means_df, x=~(t*cos(plotX_means)), y=~(t*sin(plotX_means)), z=0, type='scatter3d', showlegend=FALSE, mode='lines', name="Expected Obs X", line=list(color='green', width=15), opacity=0.8) %>% add_trace(data=means_df, x=~(t*cos(plotY_means)), y=~(t*sin(plotY_means)), z=0-epsilon, type='scatter3d', showlegend=FALSE, mode='lines', name="Expected Obs X", line=list(color='blue', width=15), opacity=0.8) %>% add_trace(data=means_df, x=~(t*cos(plotZ_means)), y=~(t*sin(plotZ_means)), z=0+epsilon, type='scatter3d', showlegend=FALSE, mode='lines', name="Expected Obs X", line=list(color='red', width=15), opacity=0.8) %>% add_trace(x=0, y=0, z=c(-2,2), name="Space", showlegend=FALSE, line=list(color='gray', width=15), opacity=0.8) %>% layout(title="Pseudo Spacekime (1D Space, 2D Kime) Kime-Phase Sampling and Foliation", scene = list(xaxis=list(title="Kappa1"), yaxis=list(title="Kappa2"), zaxis=list(title="Space"))) %>% hide_colorbar() pl_scene ```