Introduction to the post: 

This post is an attempt to introduce spacetime geometry by first explaining the significance of some invariant quantities, showing how it necessarily has some of the properties that it has as a consequence of these invariants, and then deriving the form of the Lorentz transformations which take spacetime from one observer's reference frame to another.  

This approach is intended to make it clear why the invariants are direct consequences of a few, well justified assumptions of physics, instead of the more common approach involving first deriving the form of the Lorentz transformations within a particular coordinate system.   

This post is intended to be read by anyone who wants to learn Special Relativity. 

If you have read the entire post, you may find that it is longer than it needs to be to convey its content, contains mistakes, or is not as clear as it might be. ^[1] Please do not downvote it for any of these reasons, as they are not inherently negative, instead being absences of positive qualities which would add value.  This, as well as other factors, placed pressure on me to complete the post relatively fast, which may account for some of the above deficiencies and is why I am likely to append to and amend it in the future. I would like to write an introduction to hyperbolic geometry at some point, and the hyperboloid described in this post could be a convenient manifestation of it through which to understand some of its properties. 

The term photon is used below, but as I do not understand quantum mechanics, I am not sure if it is appropriate (sorry for this). Please interpret it as a classical, point-like particle(even if this isn't actually what it is)  .

Introduction to the spacetime of Special Relativity :

A well known thought experiment involving sticks and paintbrushes shows that the dimensions of space perpendicular to the relative motion of two observers are not affected by the transformation of space-time ( a Lorentz boost, but that it takes the form associated with that name has not been established) required to go from one of their reference frames to the other. The thought experiment assumes that there are two poles, each of which has the same length in its own reference frame, and whose centres both lie on a straight line along which both are moving towards one another(for the purposes of this thought experiment, we can assume that the poles can pass through one another while still being 'paintable' , or alternatively that the paintbrushes are replaced with blades which would each slice through the opposite pole if they collided, with the aligned sections of the poles disintegrating on impact ) . If either of the two poles moving towards one another in a direction perpendicular to themselves was longer than the other in its own reference frame, then the paintbrush at the end of the other would mark it at some point along its length short of the end, while the paintbrush at its end would not make contact with the other pole. If this was the case in the reference frame of one of the two poles, then, assuming that physics is symmetrical, the opposite would have to be true in the other pole's reference frame, leading to two different events. 

However, it is not only demonstrated by observation of the actual universe that this does not seem to happen  (that events occur objectively, at least in non-quantum physics) , but it would also destroy, or at least radically alter the structure of causality itself, which underlies a vast amount of the reason why it is possible to make predictions or why time is perceived as different from space.  

Things which are left alone by transformations to go from one observer's view of the universe to another, invariants, are extremely important because they are objective properties of physics which do not depend on the observers. Unfortunately, there may be other observers moving in different directions relative to any one of them, so which directions are perpendicular is itself observer-dependent. Luckily, by assuming that events themselves are not, or at least that they occur within an objective causal structure, we can conclude that another quantity is globally invariant. Imagine a spacecraft containing a clock which is accurate in its own reference frame, for example the light clock described below, which travels from one space-station to another and has been programmed to send a signal only when it has traveled for a particular time since leaving the first, namely the time required for it to reach the other space station according to its clock, so that it will send its signal only when it has docked with the second space station. Then in order for other observers to agree on whether it sends its signal while the second space station can receive it, and therefore on the causal structure of everything consequent to this, they must also agree on the number of ticks generated by the spacecraft's clock, and therefore on the elapsed time within the spacecraft, its proper time.

Knowing this, along with the invariance of spacetime volume, it's possible to deduce a surprising amount about flat spacetime.

But why would space time volume be invariant? To explain this, it's necessary to first explain two other famous thought experiments which demonstrate two of the best known phenomena of Special Relativity, time dilation and length contraction. To demonstrate time dilation, a clock consisting of two parallel (perhaps horizontal) mirrors and a pulse of light bouncing or reflecting directly (vertically)  between them at constant time intervals is considered. Within the rest frame of this clock, the time which elapses between ticks is clearly the distance between the mirrors divided by the speed of light, which is assumed to be constant as implied by Maxwell's equations. However, in the reference frame of another observer moving parallel to the mirrors, the mirrors, and therefore the points at which the light pulse repeatedly reflects off them are moving themselves in the opposite (also horizontal) direction, and because the perpendicular distance between the two mirrors is preserved, the light clearly has to travel a longer distance than in the original reference frame, along the hypotenuse of a right-angled triangle whose other two edges stretch along (horizontally) and between (vertically) the mirrors. If the speed of light is constant, this implies that the duration of each tick must be longer in this reference frame than it was in that of the light-clock by the same factor, known as γ,  which can be calculated with Pythagoras' theorem. If we imagine one of the mirrors to be infinitely long and fixed in the latter reference frame, so that it still forms a working light clock along with the other which glides along beside(above) it, then it becomes apparent that, in either reference frame, the duration of each tick is precisely half the time taken for the smaller mirror to travel the distance between the points at which the classical photon bounces off the larger one. Given that observers in both reference frames agree on their relative speed, they must therefore disagree on these distances as they disagree on the time one of them takes to cover them; in particular, an observer in the reference frame of the smaller mirror must observe these distances to be shorter than they appear in the other frame by the same γ factor by which the durations of their ticks are shorter.

 In 3 dimensional space, it is helpful to describe, or even define, volume in terms of cubes, 3-D measure polytopes, which are 3 dimensional shapes whose edges are all either perpendicular or parallel in such a way that their volume can be calculated by multiplying the lengths of their perpendicular edges. The volume of other shapes can then be obtained by filling them with (possibly infinitely) many non-overlapping cubes and summing the individual edge-length-products of these cubes. A cube can easily be generalized to obtain a hypercube, or 4 dimensional measure polytope by introducing a 4th orthogonal direction, and 4-volumes can be defined in an analogous way in terms of products of the lengths of perpendicular edges of hypercubes. One way to do this is to 'extrude' a cube in a direction perpendicular to all of its edges, just as a square can be 'extruded' into the dimension orthogonal to it to form a cube; in space-time, there is clearly no spatial direction orthogonal to a 3-cube, so it must be extended in time, which is to say allowed to exist for a duration equivalent to its edge length. But by which factor can distances in space be converted into durations, or 'temporal distances' ? Speeds are conveniently rates of change of distance with respect to time, ratios of infinitesimal intervals of space to infinitesimal intervals of time, and seem to lend themselves to use as a conversion factor.  Because all observers agree on the speed of light, defining it to be this space-time conversion factor allows them all to agree that the distance between two mirrors of a light clock is equivalent to the duration of its ticks, even if they disagree on both.  In the same way that cubes can be used to measure 3 dimensional volumes, hypercubes can be used to measure 4-volumes in spacetime, so establishing that the 4-volume of a hypercube is an invariant should be sufficient to demonstrate the same of all volumes. To an observer in whose reference frame a hypercube exists in the way described above, it instantly flashes into existence, occupies the same cubic volume of space for one light- edge -length of time, and then vanishes. What about the perspective of an observer moving relative to the hypercube? For simplicity, we can assume that they approach the 3-dimensional cube that is its spacelike cross section in a direction parallel to some of its edges. In fact, this observer will not agree on the simultaneity of the events at which different parts of the cube begin to exist; this phenomenon is elucidated in the thought experiment involving a moving train which passes a platform in whose reference frame two points at either end are simultaneously struck by lightning. In this case we can imagine instead that in the reference frame in which the hypercube was defined two flashes of light are simultaneously coincident with the centers of the front and back faces of its cubic spatial surface, towards which the other observer is moving. Clearly, they will see the light emanating from the nearer of the two faces before they become aware of the other flash (it is helpful to assume here that the cube is transparent).  Even if they take into account the fact that it takes the latter longer to propagate through the cube to reach them, they will conclude that the nearer flash occurred later, because while the difference between the distances the light needed to travel would for them be smaller, due to length contraction, fewer ticks of their clock would elapse in this period due to time dilation, precisely cancelling out the effect of length contraction. This would mean that the ratio of the distance between the points where the flashes were emitted to the time between them being received would be the same as in the reference frame of the cube, in which the observer was moving towards the light, which could only be reconciled with the simultaneity of the two flashes in their reference frame by assuming that the speed of light was faster than it actually is. On the other hand, if we imagine that every point on a square cross-section of the cube orthogonal to its direction of motion flashed at the point in time when it came into existence within its own reference frame, we can single out any two of these points. It would take the same amount of time for light from each of them to reach a point in space half way between them in the cube's rest frame, and because distances within this plane are preserved, the approaching observer would also measure these two flashes to be simultaneous.  As they observe any two events on the slice as it comes into existence to be simultaneous, they would therefore observe the entire slice of the cube to appear in one go, after correcting for the same error due to the time taken for light to reach them. The cube would appear to them gradually, with its most distant face coming into existence first and continuously being 'extruded' into the third dimension until its front face existed. This means that if the cube was divided into many thin slices perpendicular to the direction in which the moving observer was approaching it, each slice would appear to that observer almost instantaneously. In the limit in which the number of slices into which the cube is divided approaches infinity and the slices become infinitesimally thin, the time taken for each slice to 'extrude' itself into existence also ultimately vanishes and the process becomes equivalent to one in which each slice instantly springs into existence. The same process would seem to occur in reverse to the observer as the cube vanishes, only with the slices vanishing in the same order as they appeared. The 4-volume of the hypercube in this observer's reference frame can now be obtained by summing, or in the limit integrating that of the spacetime extension of each slice.

Since each slice lives for a period longer than its lifespan in the original reference frame by the same factor γ by which its thickness and spatial volume are smaller, its 4-volume is unchanged. This implies that the whole hypercube's 4-volume is also preserved, and therefore that any 4-volume is. 

Given that only two dimensions, time and the dimension of relative motion of two observers, are affected by the transformation of spacetime from one reference frame to  another, a 2-dimensional spacetime diagram is sufficient to capture most of the interesting properties and invariants of these transformations. Space and time are usually represented as orthogonal on 2-dimensional spacetime diagrams.

Why is this?

 Are they orthogonal in reality? There is a good reason why they are which is not only a matter of definition: in the 2 and 3 dimensional Euclidean spaces with which we are familiar, transformations which preserve distances, isometries, such as rotations, also necessarily preserve areas and volumes, but only because the edges of squares and cubes used to define volumes remain orthogonal. This in turn is a consequence of the fact that an angle is simply a ratio of two distances (the length of an arc of a circle to its radius) and so is preserved when distances are. Because proper time is preserved by the transformations between reference frames, it provides a notion of distance which makes them isometries of spacetime. One reason to believe that proper time actually is spacetime distance is because the same kinds of considerations of physics which show it to be invariant also demonstrate that space-time volumes are invariant when one of the dimensions used to define them was the lifespan, which is to say proper duration, of a cube in its own reference frame. In other words, if proper time is a kind of distance in spacetime, then, provided that proper length has a similar status, it can be derived mathematically that spacetime volumes are invariant, which is a prediction which can be made without the introduction of proper time in the first place, suggesting it really is a 'true spacetime distance'. This explains why spacetime angles are preserved by a Lorentz boost,  but not necessarily why space and time need to be orthogonal in any reference frame in the first place. Perhaps the best reason is simply that it is the essence of orthogonality that two orthogonal shapes/directions share no component in common, and is seems clear that space is a separate entity from time for any one observer. Another reason is that it takes more information to specify a universe in which they are not orthogonal than one in which they are, and therefore that it is much more likely that we live in a universe in which they are.

Using the convention that the speed of light is equal to 1, we can see that the path of a single photon must be depicted as a diagonal line making an angle of with either of the axes in such a spacetime diagram. Because the transformation effected by acceleration in the spatial direction represented in the diagram preserves the speed of light, the corresponding transformation of the diagram must permute these lines among themselves. It is also clear that observers agree on whether a particular light ray is travelling towards or away from them , assuming their spatial orientations are the same, which implies that the lines of gradient 1 and -1 must be mapped to lines with the same gradients.  If the transformation represents a transition between the reference frames of two observers which meet at an event at which they synchronize their clocks and measuring rods, as is customary, then the Euclidean representation thereof must also preserve its origin. This means that the line representing the path taken by a photon emitted at this event must be transformed into itself, although this does not preclude it being stretched in the diagonal (representing lightlike or nulllike) directions away from the origin. In fact, it necessarily is stretched in this way, because if it was not, then, symmetrically, light rays travelling in the opposite direction would also not be represented as stretched, and as the transformation preserves the (infinite) angles, or alternatively the absolute speed of light at each point where any of these lightlike lines intersect, it would be possible to show that a grid consisting of squares offset by half a right angle relative to the coordinate grid of the original Cartesian coordinate system would be preserved itself, which could function as a separate coordinate system to describe spacetime, showing it to be unchanged by the transformation. The grid points in this new coordinate system would represent the events at which light rays intersect with one another in the relevant 2-dimensional slice of spacetime.  We can now see that, in order for spacetime volumes, and therefore the 2 dimensional space-time cross sectional areas of the relevant shapes in our spacetime diagram to be preserved, in particular the grid cells of this new lightlike coordinate system which appear as rotated squares, the factor by which one of the two families of diagonal lines is stretched must be the reciprocal of the factor by which the other is stretched, or equivalently, the factor by which the other family is compressed.  You might have guessed that this factor is the Lorentz factor, γ,  but sadly it is not; it is in fact $\gamma +\sqrt{{\gamma }^2-1}$ ,  as can be ascertained from observing what form time dilation and length contraction take within the spacetime diagram. 

It is now apparent that the transformation of spacetime undergone by an observer when accelerating consists of stretching spacetime in the direction of light rays travelling in the same direction through space as the observer is accelerating, and compression of spacetime in the direction of light rays travelling in what is spatially the opposite direction, which is sometimes referred to as a squeeze-mapping. Because it is an isometry which preserves a singular fixed point about which the rest of the space(time) surrounding it is transformed in a continuous way, which allows it to be broken down into infinitesimal transformations of the same kind, this transformation, known as a Lorentz boost, is the analogue of a rotation of Euclidean space. In Euclidean space, such a transformation preserves circles and spheres because they consist of all points at a particular distance from the center of rotation, and for the same reason, analogues of these shapes are preserved by Lorentz transformations in spacetime. What are they?

In the 2 - dimensional case, we can infer that in either Euclidean space or Spacetime, these curves are the only ones preserved by a rotation, because any point continuously traces out such a path as it is rotated, and for a curve to be preserved its points would need to be permuted among themselves, which would require the locus of each one of them to be contained by the curve in question, but this can only happen when it either is the locus of each of its points, or a 1-parameter family of them, which would then not be a curve.  We can find out which curves are preserved by these transformations in a 2 dimensional spacetime by examining our spacetime diagram; if they are preserved in said spacetime, their representations must also be preserved.  In Euclidean space, the curves preserved by an transformation which stretches and compresses them in orthogonal directions in such a way as to preserve areas are clearly represented in a Cartesian coordinate system whose axes are parallel to the directions in question by a curve given by the equation $xy=a$, where a represents the area of a rectangle one of whose vertices lies at the origin of the coordinate system, the opposite vertex of which lies at the point with coordinates x , y,  and whose edges are parallel to the x and y axes. The equation can be interpreted as saying " the area of any such rectangle which meets the curve is a constant" , and therefore transformations preserving these rectangles necessarily preserve its truth value.  This curve is a Hyperbola, which can alternatively be obtained by slicing a cone parallel to its axis. 

 We can understand why this is by increasing the number of spatial dimensions represented by our spacetime diagram to 2, making it 3-dimensional. There are now an infinite number of spatial directions in which light can propagate accommodated by our diagram, forming a circle, and accordingly, their representation in the spacetime diagram becomes a cone whose curved surface is inclined at 45 degrees to its vertical , temporal axis. Each horizontal cross section of the cone represents a circular wavefront of light in the two dimensions of space which the diagram now captures, emanating from a single point where a flash occurred. Although the individual lines representing light rays within this cone may be moved around by a Lorentz boost, the cone itself remains the surface representing all points reached by light travelling at its universal speed away from this one event on which all observers agree, and is therefore unchanged.  ^[2] It is also apparent that, because the additional dimension now incorporated into the diagram which is orthogonal to the direction of relative acceleration in space is unaffected by the Lorentz boost, like the cone, the 2 dimensional spacetime slices consisting of all events represented as vertical planes in the diagram are transformed into themselves, and therefore that the same is true of its curve of intersection with the cone, which we can now see must be a hyperbola. Clearly the hyperbola is infinite in length, so these rotations, known as hyperbolic rotations, must be quite unlike the rotations of Euclidean space in that they allow an object to rotate through an infinite angle without ever returning to its starting orientation. These angles are known as hyperbolic angles, or, within the context of two observers whose trajectories pass through one another at a particular event, as rapidities. Of course, in reality the light cone is a 3-dimensional surface in a 4-dimensional space, as is spatial cross sections are expanding spherical light wavefronts which are themselves 2-dimensional. I relied upon the assumption that the lightcone was itself preserved, or equivalently, that the speed of light in any direction was unaffected by a Lorentz boost, but we have not yet shown that this is possible. However, we have only defined the speed of light in the direction of relative motion of the observers, in the case of the spacetime diagram, and the speed of light perpendicular to the direction of relative motion in the case of the light clock, to be preserved.  Luckily, the light cone is indeed preserved by such a transformation because its cross-sections are elliptical, a visual explanation of which is provided by "Dandelin spheres".  ^[3] While we (probably, I do) lack the capacity to visualize a 3 dimensional surface embedded in a 4-dimensional space, we can generalize the above line of thought to conclude that a 3-dimensional space - time slice taken through the true light cone yields a surface which is also invariant and is the Space-time analogue of a 2-sphere, the familiar 2-dimensional surface of a ball in 3-dimensional Euclidean space. In a similar way to how the 2- sphere can be obtained by taking a 3-dimensional slice through a 4-dimensional 3-sphere, a 2-dimensional Hyperboloid can be obtained by slicing through a 3-dimensional Hyperboloid in spacetime, which is the surface consisting of all events in spacetime which can be reached by observers travelling for a particular amount of their own time after diverging in different directions and at different speeds slower than that of light from a particular event, or in other words, the set of all points at a particular distance from a given one in spacetime.    

The intrinsic geometry of the hyperboloid has properties which it shares with the sphere because of its rotational symmetry, known as homogeneity and isotropy.

Like a sphere, a hyperboloid, in any positive, whole  number of dimensions must be homogenous because those aspects of its geometry which are determined by distances within it are preserved by rotations, and it turns out these aspects are all there is to its intrinsic geometry!  This is because angles are defined in terms of distances, and information about distances measured along the grid-lines of a coordinate system, along with the angles between them, are sufficient to uniquely determine the geometry described by the coordinate system.  Unfortunately, however, it is not possible for an observer in special relativity to traverse one of these hyperboloids because this would require exceeding the speed of light. Within our spacetime diagram, we can see that as the tangent to an initially vertical line, representing a stationary observer in the reference frame it describes, rotates along a hyperbola, the speed of the corresponding observer approaches the speed of light but never reaches it.  In the reference frame of the spacetime diagram, its light clock would appear to tick slower and slower as the direction of the light reflected between the mirrors became closer and closer to being parallel to its direction of motion, meaning that actually reaching the speed of light would require it to stop ticking all together, which suggests that the proper time separating any two events connected by a potential light ray is in fact 0, in contrast to the fact that in Euclidean space, two points being separated by a distance of 0 would imply they were the same point.  How can we measure spacetime distances between points separated by more space than time? Clearly, in the special case in which they are coincident in our own reference frame, we can just use their spatial distance, and in the same way that proper time is invariant while time measured by another observer is not, distance measured by an observer for whom they are simultaneous is also an invariant. We can obtain a similar breakdown of causality by imagining that proper distance varies depending upon which observer views events at which light rays emanating from the centre of a sphere reach various points on its surface. ^[4] Alternatively, we can take advantage of the symmetry between spacelike and timelike directions described further on in this post. This allows us to define hyperbolic angles in terms of distances in direct analogy to the way circular angles are defined in Euclidean space, without reference to the fact that Hyperbolic rotations take the form of a squeeze-mapping. 

Isotropy refers to the rotational symmetry that a space has about particular points, which, in the case of a hyperboloid in spacetime, corresponds to symmetry with respect to rotations of space, which clearly leave time and spatial distances unchanged, and preserve the hyperboloid. Given these properties alone, it is not clear that the intrinsic geometry of the Hyperboloid is not Euclidean. In Euclidean space, it seems intuitively obvious that the curvature of a 2-sphere prevents it from containing universally parallel lines, and a similar phenomenon occurs in hyperbolic space; in particular, the intrinsic straight lines or geodesics of the sphere are great circles, whose radius is the same as that of the sphere itself, which can be obtained by slicing through the sphere with one of its planes of symmetry. This is because, if a great circles had intrinsic curvature, it would deviate to one side of the plane, but this would break the symmetry, creating a contradiction. Picture the circles generated at the intersection of a family of planes all of which intersect one another along a straight line between two opposite points on the sphere.   These all diverge from one another at one of these points, become instantaneously parallel to one another both within the sphere and within the ambient 3-dimensional Euclidean space as they pass through a further 'equatorial'  great circle, and then converge again at the opposite 'pole', meeting one another at the same angles at which they departed. The fact that instantaneously parallel lines have a tendency to converge on the sphere is referred to as a consequence of having positive Gaussian curvature, and we can see from an equivalent construction in 3-dimensional spacetime (with the aid of a corresponding spacetime diagram) that they diverge to infinity in the 2 -hyperboloid. Of course, this in itself is in accordance with the behaviour of lines in Euclidean space, but upon closer inspection, we can observe that the lines are not only moving away from one another, but actually accelerating apart.  To understand this, it is helpful to understand how an actual observer accelerates at a constant rate through spacetime. What kind of trajectory does it follow? In Newtonian physics, observers accelerating parallel to their current trajectory move in straight lines, but this is not possible in spacetime for the simple reason that it causes speed to increase, while time always flows at the same rate of 1 second per second, and as it is the appropriate kind of spacetime distance, the rate of flow of (proper) time with respect to (proper) time is the observer's speed through spacetime.  Objects accelerating in a direction perpendicular to that of their current velocity travel in circles in Newtonian physics, because a circle is a curve of constant curvature, which is to say that the rate of change of direction with respect to distance along its length does not vary. This is because of the circle's rotational symmetry, so a curve with the equivalent symmetry in spacetime must be the shape of a uniformly accelerating observer's path through spacetime. We know that this curve is a hyperbola, but it is not the spacelike hyperbola previously referred to, instead being a timelike one which is obtained by reflecting it in a light-like line. This curve is also preserved by a boost, as it is to proper length as the original is to proper time (alternatively, we can see from a 2-dimensional spacetime diagram that it is preserved because of the symmetry of the effect that a squeeze-mapping has on it). 

This same reflectional symmetry tells us that proper distance measured within our spacetime diagram appear to be spaced out at progressively greater intervals, just as the ticks of the clock of the reflected accelerating observer seemed to get slower and slower as they approached the speed of light. In contrast to this, the intrinsic circles centred at the point at which the geodesics in our Hyperboloid diverge are represented with perfect accuracy in the spacetime diagram. If the above described effect of 'length dilation' were absent, this would mean that, in the limit, as the circles grew to infinity and the hyperboloid approached tangency with the light cone it lies within, the circumferences of these circles and therefore the distances between the geodesics would grow as a linear function of their length, as they would if they actually lay on a cone. Taking the 'length dilation' into account, it is apparent that the rate at which these distances increase itself increases without bound, in a way which is the opposite of the behaviour of great circles on a sphere. 

In Euclidean space, it is helpful to define a Cartesian coordinate system to describe geometry.  If we define a similar coordinate system whose time axis is measured by the clock of some observer, running vertically up a spacetime diagram, and whose other 3 axes are inherited directly from the Euclidean space, we can use it to calculate distances. This can be done using an analogue of Pythagoras' theorem; in Euclidean space, Pythagoras' theorem can be proven by observing that a right angled trangle can be split into two smaller, similar ones by drawing a line from its right angle to its hypotenuse, and then observing that the area of each of the two smaller triangles which result is exactly the same proportion of that of the square on each one of them , as the larger triangle's area is of the area of the square on its hypotenuse. Because the sum of the areas of the two smaller triangles is that of the original larger one, the same is true of the squares, which is what the theorem states. In spacetime, it is possible for three further kinds of right angled triangles to exist, with either 1,2 or 3 sides which are timelike. For the purposes of measuring timelike distances, it is sufficient to focus on triangles with a timelike hypotenuse and one other timelike edge.  If we attempt to generalize the proof of pythagoras' theorem above, we run into the problem that an orthogonal spacelike line drawn from the hypotenuse does not intersect the right angle, which can be ascertained from a spacetime diagram. However, extending the hypotenuse beyond its end point, we can connect it to the right angle by an orthogonal line. ^[5] Contrary to the Euclidean case, this would adjoin a new triangle to the original one to produce a third one containing it.

As in Euclidean geometry, both of these triangles would be similar to the original one, as the larger of them would share one vertex with the original, (I would suggest drawing a spacetime diagram to verify this), but while in the Euclidean theorem the areas, and therefore the squares associated with each of these triangles would need to be added together to give that of the original triangle, here they are subtracted. 

In other words, the square on the timelike hypotenuse is equal to the difference between the squares on the other timelike side and the spacelike side.  This produces the following extremely important expression for the spacetime distance between two events, at least when they are timelike- separated:   

$\Delta {\tau }^2=\Delta t^2-l^2=\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$ , where t is the length of the timelike edge of the triangle, τ is the length of the hypotenuse, and l is the length of the spacelike edge of the triangle.

This formula can also be written as

$d{\tau }^2=d{t}^2-d{x}^2-d{y}^2-d{z}^2$ or $\tau =\sqrt{d{t}^2-d{x}^2-d{y}^2-d{z}^2}$ , where the coordinate intervals are now infinitesimal, allowing the formula to be extended to cover all space-times which resemble that of special relativity on an infinitesimal scale, including the curved spacetime of General relativity.

This is known as the metric, and it is often considered to generate the structure of much of the geometry of spacetime. For example, the light cone emerges as the surface consisting of all events for which the metric dictates that the spacetime distance to a given event vanishes. As almost every component of this kind of geometry can be defined in terms of distance (apart from the number of dimensions, which is also contained in the metric),  knowing what it is is sufficient to recover the entire structure of spacetime.  We can see that when we allow the spacelike edge of the triangle to exceed the length of the longer timelike one, the formula for the distance becomes an imaginary number,  $i\sqrt{d{x}^2+d{y}^2+d{z}^2-d{t}^2}$ , but by the symmetry between time and space, this is simply imaginary proper length.  Written this way, but without the i, the metric is expressed in a spacelike form, and is referred to as having a -+++ signature as opposed to +--- .

1. ^{^}

   I have attempted to introduce terminology throughout the post, so that a reader new to it might be able to recognize it when reading further into the subject. Unfortunately, this requires me to describe each of the new terms as I introduce them, increasing the length of the post and making it less interesting to read, or alternatively use them in such a way that I hope the context will make it clear what they mean, which may lead to confusion. I might also have neglected to explain certain things. Please only downvote the post because of them if the rest of it is contingent on them.

2. ^{^}

    Physically, we are justified by the form of Maxwell's equations in making this assumption, but we have not yet shown that this is possible. Mathematically, we have only defined the speed of light in the direction of relative motion of the observers, in the case of the spacetime diagram, and the speed of light perpendicular to the direction of relative motion in the case of the light clock, to be preserved.  Luckily, as mentioned in the main body of the text, the light cone is indeed preserved by such a transformation because its cross-sections are elliptical, a visual explanation of which is provided by "Dandelin spheres". 

3. ^{^}

   Alternatively, we can establish that this is the case as a consequence of some properties that hyperbolic rotations share with their Euclidean counterparts. One of these is that they form a continuous family and may be composed with one another. This means that the result of a series of successive hyperbolic rotations through a particular angle is to rotate through the sum of the hyperbolic angles of each of them. As we have defined the Lorentz boost to preserve the speed of a light ray perpendicular to the direction of acceleration( and therefore to the plane of rotation), we know that it will remain within the light cone under a hyperbolic rotation through any hyperbolic angle. In addition, any light ray (which you are encouraged to visualize as a diagonal line along the surface of the cone in a spacetime diagram) can be forced to become perpendicular to this direction by applying the inverse of the transformation which would take a perpendicular ray to it. In analogy to Euclidean rotations, an inverse of any hyperbolic rotation always exists as one through the negative of the original (hyperbolic) angle. From within the physical universe, this corresponds to accelerating to catch up with a light clock moving past the observer in a direction parallel to its mirrors. Accelerating further in the same direction(or the opposite one) takes the direction(s) of the photon bouncing between the mirrors anywhere within the 2-dimensional light cone which is represented by the spacetime diagram apart from either of the two directions parallel with the acceleration.  The combined effect of these two Lorentz boosts is to transform any light ray within this 2-dimensional light cone to another, and is equivalent to a single boost in which the observer's own velocity caught up with, or decelerated to meet, and then accelerated(or decelerated) past that of the light clock to reach its new reference frame. Because every Lorentz boost takes this form with respect to any light ray within the above 2-dimensional light cone apart from those spatially parallel to the acceleration, this light cone is indeed preserved. Applying the same logic to light rays travelling in other directions in space shows that the true, 3-dimensional light cone is also preserved along with the speed of light in any direction.

4. ^{^}

   I am not completely confident that this thought experiment has the desired conclusion without relying on facts that haven't yet been demonstrated; if you can think of a way to improve it, please comment. 

5. ^{^}

   Within spacetime, this would appear in the form of two points in space which spring into existence simultaneously, one of which moves away from the other at a constant speed less than that of light before both of them vanish, simultaneously in the reference frame of one of them, at which instant a (purely spatial) line exists in the same reference frame connecting the spatial endpoints of their paths. The worldline of the point in whose reference frame this is the case would be the longer timelike edge of the triangle due to time dilation, but it is in fact not the hypotenuse, because it is clearly orthogonal to the spacelike edge. From the reference frame of the other point, whose worldline was the hypotenuse of the triangle, the spacelike edge would not come into existence instantly. It would instead appear to 'extend' itself into, and simultaneously out of existence, as a point moving along its length faster than light. The point would begin life at the event at the end of the hypotenuse, and progress towards the end of the other timelike edge, which is to say the event at which the other point moving in this reference frame vanished, and vanish itself at the same moment in time. From the reference frame associated with the hypotenuse, we can see that a perpendicular line could not be drawn to the vertex with the equivalent of a right angle, because the hypotenuse would already have ceased to exist; nonetheless, extending the hypotenuse further, or , in other words, prolonging the lifespan of its spatial point- cross section, a new, larger right angled triangle can be formed by waiting until the moment at which the other two edges vanish to introduce a purely spacelike side in this reference frame.