diff --git a/docs/src/newsreport.md b/docs/src/newsreport.md
index 29e9b24..8705b19 100644
--- a/docs/src/newsreport.md
+++ b/docs/src/newsreport.md
@@ -305,6 +305,8 @@ This is related to the fact that the weak interaction in the Standard Model is n
 
 ![35](./assets/newsreport/35.PNG)
 
+[The tangent bundle and the cotangent bundle](https://github.com/iamazadi/Porta.jl/blob/master/models/newsreport/gaugeconnections/fig1512cotangentbundle.jl)
+
 ![36](./assets/newsreport/36.PNG)
 
 ![37](./assets/newsreport/37.PNG)
diff --git a/models/newsreport/gaugeconnections/fig1512cotangentbundle.jl b/models/newsreport/gaugeconnections/fig1512cotangentbundle.jl
new file mode 100644
index 0000000..7229167
--- /dev/null
+++ b/models/newsreport/gaugeconnections/fig1512cotangentbundle.jl
@@ -0,0 +1,407 @@
+using FileIO
+using GLMakie
+using Porta
+
+
+figuresize = (4096, 2160)
+segments = 360
+frames_number = 360
+modelname = "fig1512cotangentbundle"
+x̂ = ℝ³([1.0; 0.0; 0.0])
+ŷ = ℝ³([0.0; 1.0; 0.0])
+ẑ = ℝ³([0.0; 0.0; 1.0])
+eyeposition = normalize(ℝ³(1.0, 1.0, 1.0)) * float(π)
+lookat = ℝ³(0.0, 0.0, 0.0)
+up = normalize(ℝ³(0.0, 0.0, 1.0))
+sphereradius = 1.0
+mask = load("data/basemap_mask.png")
+reference = load("data/basemap_color.png")
+attributespath = "data/naturalearth/geometry-attributes.csv"
+nodespath = "data/naturalearth/geometry-nodes.csv"
+boundary_names = Set()
+boundary_nodes = Vector{Vector{ℝ³}}()
+points = Vector{Vector{ℍ}}()
+indices = Dict()
+T, X, Y, Z = vec(normalize(ℝ⁴(1.0, 0.0, 1.0, 0.0)))
+u = 𝕍(T, X, Y, Z)
+q = ℍ(T, X, Y, Z)
+tolerance = 1e-3
+@assert(isnull(u, atol = tolerance), "u in not a null vector, $u.")
+@assert(isapprox(norm(q), 1, atol = tolerance), "q in not a unit quaternion, $(norm(q)).")
+ϵ = 0.1
+timesign = 1
+T = float(timesign)
+gauge1 = 0.0
+gauge2 = π / 2
+gauge3 = float(π)
+gauge4 = 3π / 2
+gauge5 = 2π
+latitudescale = 1 / 2
+longitudescale = 1 / 4
+chart = (-π * latitudescale / 2, π * latitudescale / 2, -π * longitudescale, π * longitudescale)
+M = Identity(4)
+markersize = 0.05
+linewidth = 20
+arrowsize = Vec3f(0.06, 0.08, 0.1)
+arrowlinewidth = 0.04
+arrowscale = 0.5
+fontsize = 0.25
+zero = Point3f(0.0, 0.0, 0.0)
+lspace1 = range(-π, stop = float(π), length = segments)
+lspace2 = range(-π / 2, stop = π / 2, length = segments)
+names1 = ["q₁", "q₂", "q₃", "q₄", "q₅", "q₆"]
+names2 = ["q"]
+colorants1 = [:black for _ in eachindex(names1)]
+colorants2 = [:black for _ in eachindex(names2)]
+colormaps1 = [:rainbow for _ in eachindex(names1)]
+colormaps2 = [:rainbow for _ in eachindex(names2)]
+number1 = length(names1) # the number of tangent bundles
+number2 = length(names2) # the number of tangent bundles
+tangentbundles1 = TangentBundle[]
+tangentbundles2 = TangentBundle[]
+pathsegments = frames_number
+
+makefigure() = Figure(size = figuresize)
+fig = with_theme(makefigure, theme_black())
+pl = PointLight(Point3f(0), RGBf(0.0862, 0.0862, 0.0862))
+al = AmbientLight(RGBf(0.9, 0.9, 0.9))
+lscene1 = LScene(fig[1, 1], show_axis=true, scenekw = (lights = [pl, al], clear=true, backgroundcolor = :white))
+lscene2 = LScene(fig[1, 2], show_axis=true, scenekw = (lights = [pl, al], clear=true, backgroundcolor = :white))
+
+## Load the Natural Earth data
+countries = loadcountries(attributespath, nodespath)
+while length(boundary_names) < 10
+    push!(boundary_names, rand(countries["name"]))
+end
+for i in eachindex(countries["name"])
+    for name in boundary_names
+        if countries["name"][i] == name
+            push!(boundary_nodes, countries["nodes"][i])
+            println(name)
+            indices[name] = length(boundary_nodes)
+        end
+    end
+end
+for i in eachindex(boundary_nodes)
+    _points = Vector{ℍ}()
+    for node in boundary_nodes[i]
+        r, θ, ϕ = convert_to_geographic(node)
+        push!(_points, q * ℍ(exp(ϕ * longitudescale * K(1) + θ * latitudescale * K(2))))
+    end
+    push!(points, _points)
+end
+basemap1 = Basemap(lscene1, q, gauge1, M, chart, segments, mask, transparency = true)
+basemap2 = Basemap(lscene1, q, gauge2, M, chart, segments, mask, transparency = true)
+basemap3 = Basemap(lscene1, q, gauge3, M, chart, segments, mask, transparency = true)
+basemap4 = Basemap(lscene1, q, gauge4, M, chart, segments, mask, transparency = true)
+basemap = Basemap(lscene2, q, gauge1, M, chart, segments, mask, transparency = true)
+
+whirls1 = []
+whirls2 = []
+whirls3 = []
+whirls4 = []
+for i in eachindex(boundary_nodes)
+    color1 = getcolor(boundary_nodes[i], reference, 0.1)
+    color2 = getcolor(boundary_nodes[i], reference, 0.2)
+    color3 = getcolor(boundary_nodes[i], reference, 0.3)
+    color4 = getcolor(boundary_nodes[i], reference, 0.4)
+    whirl1 = Whirl(lscene1, points[i], gauge1, gauge2, M, segments, color1, transparency = true)
+    whirl2 = Whirl(lscene1, points[i], gauge2, gauge3, M, segments, color2, transparency = true)
+    whirl3 = Whirl(lscene1, points[i], gauge3, gauge4, M, segments, color3, transparency = true)
+    whirl4 = Whirl(lscene1, points[i], gauge4, gauge5, M, segments, color4, transparency = true)
+    push!(whirls1, whirl1)
+    push!(whirls2, whirl2)
+    push!(whirls3, whirl3)
+    push!(whirls4, whirl4)
+end
+
+for i in 1:number1
+    transparency = i == 1 ? false : true
+    tangentbundle1 = TangentBundle(lscene1, names1[i], segments = segments, pathsegments = pathsegments, transparency = transparency,
+        markersize = markersize, linewidth = linewidth, color = colorants1[i], colormap = colormaps1[i], arrowsize = arrowsize,
+        arrowlinewidth = arrowlinewidth, fontsize = fontsize)
+    push!(tangentbundles1, tangentbundle1)
+end
+for i in 1:number2
+    transparency = i == 1 ? false : true
+    tangentbundle2 = TangentBundle(lscene2, names2[i], segments = segments, pathsegments = pathsegments, transparency = transparency,
+        markersize = markersize, linewidth = linewidth, color = colorants2[i], colormap = colormaps2[i], arrowsize = arrowsize,
+        arrowlinewidth = arrowlinewidth, fontsize = fontsize)
+    push!(tangentbundles2, tangentbundle2)
+end
+
+vobservable = Observable(Point3f(x̂))
+uobservable = Observable(Point3f(ŷ))
+zobservable = Observable(Point3f(ẑ))
+aobservable = Observable("0.000 𝑖")
+head = Observable(Point3f(x̂))
+
+v2observable = Observable(Point3f(x̂))
+v3observable = Observable(Point3f(x̂))
+v4observable = Observable(Point3f(x̂))
+u2observable = Observable(Point3f(ŷ))
+u3observable = Observable(Point3f(ŷ))
+u4observable = Observable(Point3f(ŷ))
+z2observable = Observable(Point3f(ẑ))
+z3observable = Observable(Point3f(ẑ))
+z4observable = Observable(Point3f(ẑ))
+a2observable = Observable("0.000 𝑖")
+a3observable = Observable("0.000 𝑖")
+a4observable = Observable("0.000 𝑖")
+
+ps = @lift([$zobservable, $zobservable, $(tangentbundles2[1].tangenttail)])
+ns = @lift([$uobservable, $vobservable, $head])
+colorants = [:red, :green, :blue]
+arrows!(lscene2,
+    ps, ns, fxaa = true, # turn on anti-aliasing
+    color = colorants,
+    linewidth = arrowlinewidth, arrowsize = arrowsize,
+    align = :origin, transparency = false
+)
+
+ps234 = @lift([$z2observable, $z2observable, $z3observable, $z3observable, $z4observable, $z4observable])
+ns234 = @lift([$u2observable, $v2observable, $u3observable, $v3observable, $u4observable, $v4observable])
+arrows!(lscene2,
+    ps234, ns234, fxaa = true, # turn on anti-aliasing
+    color = [colorants[1:2]..., colorants[1:2]..., colorants[1:2]...],
+    linewidth = arrowlinewidth, arrowsize = arrowsize,
+    align = :origin, transparency = true
+)
+
+znobservables = Observable(Point3f[])
+vnobservables = Observable(Point3f[])
+unobservables = Observable(Point3f[])
+N = 30
+for i in 1:N
+    push!(znobservables[], Point3f(ẑ))
+    push!(vnobservables[], Point3f(ẑ))
+    push!(unobservables[], Point3f(ẑ))
+end
+psn = znobservables
+nsvn = vnobservables
+nsun = unobservables
+arrows!(lscene2,
+    psn, nsvn, fxaa = true, # turn on anti-aliasing
+    color = [colorants[2] for _ in 1:N],
+    linewidth = arrowlinewidth * arrowscale, arrowsize = arrowsize .* arrowscale,
+    align = :origin, transparency = true
+)
+arrows!(lscene2,
+    psn, nsun, fxaa = true, # turn on anti-aliasing
+    color = [colorants[1] for _ in 1:N],
+    linewidth = arrowlinewidth * arrowscale, arrowsize = arrowsize .* arrowscale,
+    align = :origin, transparency = true
+)
+
+titles = @lift(["u", "v", $aobservable, "a"])
+rotation = gettextrotation(lscene2)
+
+text!(lscene2,
+    @lift([$zobservable + $uobservable, $zobservable + $vobservable,
+        $zobservable + Point3f(normalize(ℝ³($uobservable + $vobservable))),
+        $(tangentbundles2[1].tangenttail) + $head]),
+    text = titles,
+    color = [colorants..., :blue],
+    rotation = rotation,
+    align = (:left, :baseline),
+    fontsize = fontsize,
+    markerspace = :data, transparency = false
+)
+titles234 = @lift([$a2observable, $a3observable, $a4observable])
+text!(lscene2,
+    @lift([$z2observable + Point3f(normalize(ℝ³($u2observable + $v2observable))),
+        $z3observable + Point3f(normalize(ℝ³($u3observable + $v3observable))),
+        $z4observable + Point3f(normalize(ℝ³($u4observable + $v4observable)))]),
+    text = titles234,
+    color = [:blue, :blue, :blue],
+    rotation = rotation,
+    align = (:left, :baseline),
+    fontsize = fontsize,
+    markerspace = :data, transparency = true
+)
+
+arcpoints = Observable(Point3f[])
+_arcpoints = Observable(Point3f[])
+arccolors = collect(1:segments)
+lines!(lscene2, arcpoints, color = arccolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = :prism)
+lines!(lscene2, _arcpoints, color = arccolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = :prism)
+
+arcpoints2 = Observable(Point3f[])
+arcpoints3 = Observable(Point3f[])
+arcpoints4 = Observable(Point3f[])
+lines!(lscene2, arcpoints2, color = arccolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = :prism, transparency = true)
+lines!(lscene2, arcpoints3, color = arccolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = :prism, transparency = true)
+lines!(lscene2, arcpoints4, color = arccolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = :prism, transparency = true)
+
+sectionalpath = Observable(Point3f[])
+sectionalpathcolors = collect(1:pathsegments)
+lines!(lscene2, sectionalpath, color = sectionalpathcolors, linewidth = linewidth / 2, colorrange = (1, pathsegments), colormap = :lightrainbow)
+
+
+getconnection(q::ℍ) = begin
+    x₁, x₂, x₃, x₄ = vec(q)
+    z₀ = x₁ + im * x₂
+    z₁ = x₃ + im * x₄
+    @assert(isapprox(abs(z₀)^2 + abs(z₁)^2, 1), "The point $_q is not in S³, in other words: |z₀|² + |z₁|² ≠ 1.")
+    # the infinitestimal action of U(1) on S³
+    v = ℝ⁴(vec(ℍ([im * z₀; im * z₁])))
+    # z ∈ ℂ²
+    z = ℝ⁴(x₁, x₂, x₃, x₄)
+    # u ∈ TS³
+    u = ℝ⁴(-x₂, x₁, -x₄, x₃)
+    @assert(isapprox(dot(z, u), 0), "The vector $u is not tangent to S³ at point $z. in other words: <z, u> ≠ 0.")
+    # a unique connection one-form on S³ with values in ℝ𝑖 such that ker a = v⟂
+    a = dot(v, u) * im
+    u, v, a
+end
+
+
+animate(frame::Int) = begin
+    progress = Float64(frame / frames_number)
+    println("Frame: $frame, Progress: $progress")
+    ψ = progress * π
+    θ, ϕ = cos(2ψ) * π / 2, sin(4ψ) * π
+    _q = q * ℍ(exp(ϕ * longitudescale * K(1) + θ * latitudescale * K(2)))
+    x₁, x₂, x₃, x₄ = vec(_q)
+    z₀ = x₁ + im * x₂
+    z₁ = x₃ + im * x₄
+    z = ℝ⁴(x₁, x₂, x₃, x₄)
+    u, v, a = getconnection(_q)
+    
+    zobservable[] = Point3f(project(z))
+    uobservable[] = Point3f(normalize(project(u)))
+    vobservable[] = Point3f(normalize(project(v)))
+    aobservable[] = "$(round(imag(a), digits = 3)) 𝑖"
+
+    q2 = _q * ℍ(exp(K(3) * gauge2))
+    q3 = _q * ℍ(exp(K(3) * gauge3))
+    q4 = _q * ℍ(exp(K(3) * gauge4))
+    z2observable[] = Point3f(project(normalize(M * (q2))))
+    z3observable[] = Point3f(project(normalize(M * (q3))))
+    z4observable[] = Point3f(project(normalize(M * (q4))))
+    u2, v2, a2 = getconnection(q2)
+    u3, v3, a3 = getconnection(q3)
+    u4, v4, a4 = getconnection(q4)
+    u2observable[] = Point3f(normalize(project(u2)))
+    u3observable[] = Point3f(normalize(project(u3)))
+    u4observable[] = Point3f(normalize(project(u4)))
+    v2observable[] = Point3f(normalize(project(v2)))
+    v3observable[] = Point3f(normalize(project(v3)))
+    v4observable[] = Point3f(normalize(project(v4)))
+    a2observable[] = "$(round(imag(a2), digits = 3)) 𝑖"
+    a3observable[] = "$(round(imag(a3), digits = 3)) 𝑖"
+    a4observable[] = "$(round(imag(a4), digits = 3)) 𝑖"
+    
+    for i in 1:N
+        gauge = i / N * 2π
+        qn = _q * ℍ(exp(K(3) * gauge))
+        znobservables[][i] = Point3f(project(normalize(M * (qn))))
+        un, vn, an = getconnection(qn)
+        unobservables[][i] = Point3f(normalize(project(un)) * arrowscale)
+        vnobservables[][i] = Point3f(normalize(project(vn)) * arrowscale)
+    end
+    notify(znobservables)
+    notify(vnobservables)
+    notify(unobservables)
+
+    κ = SpinVector(z₀, z₁, timesign)
+    κ′ = SpinVector(Complex(κ) - 1.0 / √2 * ϵ / κ.a[2], timesign)
+    κprojection = project(normalize(ℍ(vec( 𝕍( κ)))))
+    κ′projection = project(normalize(ℍ(vec( 𝕍( κ′)))))
+    # shift bundles one time step
+    for i in (number1 - 1):-1:1
+        tangentbundles1[i + 1].tangenthead[] = tangentbundles1[i].tangenthead[]
+        tangentbundles1[i + 1].tangenttail[] = tangentbundles1[i].tangenttail[]
+        tangentbundles1[i + 1].a[] = tangentbundles1[i].a[]
+        tangentbundles1[i + 1].b[] = tangentbundles1[i].b[]
+        tangentbundles1[i + 1].c[] = tangentbundles1[i].c[]
+        tangentbundles1[i + 1].d[] = tangentbundles1[i].d[]
+        tangentbundles1[i + 1].fiber[] = tangentbundles1[i].fiber[]
+        tangentbundles1[i + 1].tangentcircle[] = tangentbundles1[i].tangentcircle[]
+        tangentbundles1[i + 1].basepath[] = tangentbundles1[i].basepath[]
+    end
+    for i in (number2 - 1):-1:1
+        tangentbundles2[i + 1].tangenthead[] = tangentbundles2[i].tangenthead[]
+        tangentbundles2[i + 1].tangenttail[] = tangentbundles2[i].tangenttail[]
+        tangentbundles2[i + 1].a[] = tangentbundles2[i].a[]
+        tangentbundles2[i + 1].b[] = tangentbundles2[i].b[]
+        tangentbundles2[i + 1].c[] = tangentbundles2[i].c[]
+        tangentbundles2[i + 1].d[] = tangentbundles2[i].d[]
+        tangentbundles2[i + 1].fiber[] = tangentbundles2[i].fiber[]
+        tangentbundles2[i + 1].tangentcircle[] = tangentbundles2[i].tangentcircle[]
+        tangentbundles2[i + 1].basepath[] = tangentbundles2[i].basepath[]
+    end
+    tangentbundles1[1].tangenthead[] = Point3f(normalize(κ′projection - κprojection))
+    tangentbundles1[1].a[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge1))))))
+    tangentbundles1[1].b[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge2))))))
+    tangentbundles1[1].c[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge3))))))
+    tangentbundles1[1].d[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge4))))))
+    tangentbundles1[1].fiber[] = [Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * α)))))) for α in range(0, stop = 2π, length = segments)]
+    tangentbundles1[1].tangentcircle[] = Point3f[]
+    tangentbundles2[1].tangenthead[] = Point3f(normalize(κ′projection - κprojection))
+    tangentbundles2[1].a[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge1))))))
+    tangentbundles2[1].b[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge2))))))
+    tangentbundles2[1].c[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge3))))))
+    tangentbundles2[1].d[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge4))))))
+    tangentbundles2[1].fiber[] = [Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * α)))))) for α in range(0, stop = 2π, length = segments)]
+    tangentbundles2[1].tangentcircle[] = Point3f[]
+    for i in 1:segments
+        α = exp(im * i / segments * 2π)
+        κ = SpinVector(α * z₀, α * z₁, timesign)
+        κ′ = SpinVector(Complex(κ) - 1.0 / √2 * ϵ / κ.a[2], timesign)
+        pa = ℝ³(hopfmap(normalize(ℍ(vec(κ)))))
+        pb = ℝ³(hopfmap(normalize(ℍ(vec(κ′)))))
+        if i == 1
+            tangentbundles1[1].tangenttail[] = Point3f(pa)
+            push!(tangentbundles1[1].basepath[], tangentbundles2[1].tangenttail[])
+            tangentbundles2[1].tangenttail[] = Point3f(pa)
+            push!(tangentbundles2[1].basepath[], tangentbundles2[1].tangenttail[])
+        end
+        push!(tangentbundles1[1].tangentcircle[], Point3f(pa + normalize(pb - pa)))
+        push!(tangentbundles2[1].tangentcircle[], Point3f(pa + normalize(pb - pa)))
+    end
+    for i in 1:number1
+        notify(tangentbundles1[i].basepath)
+        notify(tangentbundles1[i].tangentcircle)
+    end
+    for i in 1:number2
+        notify(tangentbundles2[i].basepath)
+        notify(tangentbundles2[i].tangentcircle)
+    end
+    h = ℍ(imag(a), normalize(ℝ³(tangentbundles1[1].tangenttail[])))
+    head[] = Point3f(rotate(ℝ³(tangentbundles1[1].tangenthead[]), h))
+    arcpoints[] = [zobservable[] + Point3f(normalize(α * ℝ³(uobservable[]) + (1 - α) * ℝ³(vobservable[]))) for α in range(0, stop = 1, length = segments)]
+    arcpoints2[] = [z2observable[] + Point3f(normalize(α * ℝ³(u2observable[]) + (1 - α) * ℝ³(v2observable[]))) for α in range(0, stop = 1, length = segments)]
+    arcpoints3[] = [z3observable[] + Point3f(normalize(α * ℝ³(u3observable[]) + (1 - α) * ℝ³(v3observable[]))) for α in range(0, stop = 1, length = segments)]
+    arcpoints4[] = [z4observable[] + Point3f(normalize(α * ℝ³(u4observable[]) + (1 - α) * ℝ³(v4observable[]))) for α in range(0, stop = 1, length = segments)]
+    _arcpoints[] = [tangentbundles2[1].tangenttail[] + Point3f(normalize(α * ℝ³(tangentbundles2[1].tangenthead[])+ (1 - α) * ℝ³(head[]))) for α in range(0, stop = 1, length = segments)]
+    push!(sectionalpath[], zobservable[])
+    notify(arcpoints)
+    notify(arcpoints2)
+    notify(arcpoints3)
+    notify(arcpoints4)
+    notify(_arcpoints)
+    notify(sectionalpath)
+    global lookat =  ℝ³(tangentbundles1[1].tangenttail[])
+    global eyeposition = normalize(lookat) * float(π + π / 2)
+    updatecamera!(lscene1, eyeposition, lookat, up)
+    updatecamera!(lscene2, eyeposition, lookat, up)
+end
+
+
+animate(1)
+
+# initialize all instances of tangent bundles before recording
+for i in 1:max(number1, number2)
+    animate(1)
+end
+for i in 1:number1
+    tangentbundles1[i].basepath[] = Point3f[]
+end
+for i in 1:number2
+    tangentbundles2[i].basepath[] = Point3f[]
+    sectionalpath[] = Point3f[]
+end
+record(fig, joinpath("gallery", "$modelname.mp4"), 1:frames_number) do frame
+    animate(frame)
+end
\ No newline at end of file