Space, the aspect of placement. It is, what defines our dimensions and relative coordinates in the universe. Without space, the system's existence can't be observed and measured by standard means. Space is the cradle for matter and aether and serves as a medium that allows their systems to interact and transform each other in ways verifiable under the rules of order.

Spatial science and magic is an advanced study. The latter requires immense knowledge and control in terms of precise magic. It's usually taught in advanced classes, but we'll give a few examples that might help us to understand the basics of how space can be manipulated.

We start with the basics, which are examples of fundamental operations on space – manipulation of boundary, scaling, rotation, translation, separation and union, creation and erasure, duplication and synchronization. As usually, understanding the inner workings of each operation requires a proper rigorous approach or intuitive knowledge about the invocation. In our cases, spatial systems might be inherently connected with other physical aspects, so we'll consider the operations in relation to them.

We start with the boundary. Spatial systems don't have to be compatible or have similar properties. Such systems are separated by what we call the boundary. Depending on the system, the three basic examples of the boundary are:

Null-discrete boundary – it's a space of infinite density and infinite thinness

Infinite-discrete boundary – it's a space of zero density and infinite size, which as some of you have already realized, has a connection to limbo

Continuous boundary – it's a space with changes in density and size that can be described by a continuous function

Of course, there're other more exotic examples of boundaries, and these boundaries can be mixed. On top of that, a system doesn't have to possess boundary, or can be its own boundary. That is, boundaries can exist as separate spatial systems.

Any operation on a system is also an operation on its boundary, these are almost always not separable. One of the exceptions is operations on the boundary itself, since they can exist as a separate system. However, these operations can be restricted, depending on the system. For example, you can remove a boundary between two systems, but only if these systems are compatible with each other.

Due to the intuitive nature of casting, it's deemed very challenging, although in some cases necessary, to establish a perfect discrete boundary. One of the spells that are based on the establishment of boundary is the spatial shield, which in its simplest form is either a null-discrete slice of a plane or a sphere.

Now, the physical aspects come into play. The boundary can be either created with help of aether, or other spatial systems can be transformed into boundaries. In the first case, based on the purity of our invocation, we get a very thin solid object that separates two compatible systems. In the second case, the existing space gets compressed into the desired shape of the desired thinness. In both cases, due to the impurity of the casting, the physical objects can be broken, which would shatter the boundary. On top of that, even pure boundary is malleable, which means, under specific conditions, it can be transformed by operations. If the systems were incompatible, it can be, for example, dented or rotated, what would also affect the system inside. If the systems are compatible, even shattering or cutting of the boundary into pieces might occur.

The other use of boundary is the creation of pocket dimensions, which are allowed to have different spatial properties than those in the world of origin. To cross into such a realm, there are many methods, but for now, we'll only consider neighboring systems and the easiest solutions.

If two systems are compatible, the easiest way to cross the boundary is to remove it.

If the systems are incompatible, the easiest way to cross the boundary is a compatible instant spatial translation into a system. In simpler terms, you teleport to a system, while performing other operations, such as scaling, to become compatible with the other spatial system.

However, If two systems are incompatible and there exists a continuous boundary, that transforms the spatial properties of one system into another, you can simply cross the boundary 'by foot'. The boundary will adjust your spatial properties by itself the moment you cross it. It doesn't require knowledge of high-end magic, but you must still consider the risk linked to the passage, such as getting lost in the boundary construct.

Now, that we understand the boundary, we can study the scaling operation. It's an operation that reduces or increases the spatial size of a system, which based on our invocation, might also scale other properties of the system.

In the most basic case, we don't scale the properties of the system, but only the space. In such a case, invocation of scaling requires us to establish a boundary for the new system and the new system becomes incompatible, the boundary can be either discrete or continuous. If the boundary is removed, the two systems will change their scale proportionally to their spatial size until they both become compatible.

If we scale the properties of a system, the system remains compatible, but the requirements to cast spells become different, e.g. spells might have only a temporary effect or require more advanced invocations.

We now will give a few applications of scaling magic.

- Shrink heavy objects to transport them easily

- Enlarge light objects temporarily and throw them at the enemy

- Shrinkable and enlengthable spears and staffs

- Infinite-discrete boundary scaled to the length of a gun barrel, modified with time-acceleration, allowing bullets to reach speeds capable of destroying even nightmares

- Enlarge food

- Temporarily enlarge waterfalls in dam power plants

- Shrink many layers of armor to create durable, yet light and thin plates or clothes

Now, it's a turn for rotation. Rotation simply allows us to displace coordinate axes of the target system. It's an operation that doesn't change the compatibility of the system. It's a basic spell, way less demanding, and easier to use, than scaling.

Applications are as follows:

- Manipulation of heavy objects

- Changing path of missiles or redirection of attacks

- Bending light, invisibility

- Whirling blades

- Transportation

- Twisting limbs, breaking bones

Next, comes translation. It allows us to displace the coordinates of the target system. It's an operation that doesn't change the compatibility of the system, as long as it is performed inside a compatible system or between compatible systems. Also a basic spell, less demanding than scaling. In case our rigorous cast allows us to take instant action, this spell equals teleportation.

Applications are as follows:

- Universal travel in case we can use instant translation

- Spatial flight with non-instant translation

- Pushing or pulling objects with immense force

- Telekinesis

Next, separation and union. These are highly advanced spatial spells, requiring us to establish or remove boundaries. We start with separation, it divides the target system into a number of smaller dimensions or subsystems, establishing one fragmented boundary between them when necessary. Upon separation, resulting subsystems don't have to be compatible. Union glues together compatible systems into a single system, removing boundaries between them. In the case of union, when the systems are incompatible, it's not uncommon to force compatibility by altering other aspects of the system, such as its physical state.

It's worth noting, that in case of separation, the system can act as If it remained as a whole, with only its dimensional aspects of subsystems changed, such as position in space or rotation.

Applications are as follows:

- Quartering your enemies

- Forming abominations

- One mind with divided bodies or limbs, capable of multi-tasking and coordinating complicated tasks

- Creating bridges and portals between worlds

Next, we have creation and erasure. It allows to create a spatial system or erase it from existence. Highly advanced spells may require the construction of boundaries or might result in two worlds being proportionally adjusted to each other, in terms of all operations.

Applications are as follows:

- Removal of obstructions of faulty systems

- Creation of pocket dimensions

- Banishment

At last, we have duplication and synchronization. Both are near-end-tier types of magic. The first type of spatial magic allows us to create a clone of a given spatial system, while the other takes two spatial systems and tries to merge them together into a compatible system.

Applications are as follows:

- Creation of hive minds

- Multiple minds controlling single body system

- Replaceable body organs

- Armies of clones

Of course, all of the operations can be combined with each other, which can result in applications such as:

- Spatial attacks, i.e. one blade striking multiple enemies at different locations

- Magical storage of miniaturized objects in pocket dimensions

As homework, each student is required to give at least one new example for each mentioned category of applications.