Puella Magi and Homura Akemi [SPOILERS up to Episode 10]
Puella Magi; The Basics
Homura's Concept Art/Regular Schoolgirl Outfit.
In the series, the main characters are girls who have been contracted by a creature calling himself Kyuubey to become magical girls (or Puella Magi) in exchange for a wish of their choosing. The purpose of being a Puella Magi is to fight against dark creatures called Witches. The purpose of witches are to spread despair and misery throughout the world, through various means. Most common is imprinting a human with their mark and filling them with thoughts of suicide, until they submit to these forced feelings and kill themselves. The more misery they spread the more powerful they are. Normal humans cannot perceive these witches and are thus blind to their wickedness.
Puella Magi, in contrast, are made to spread hope throughout the land by defeating these creatures. Not everyone is capable of becoming a Puella Magi - in episode 11 Kyuubey stated that the potential of a girl that becomes one is based on "how much despair she will face in the future". So someone who is destined to live in grief and sadness will become a stronger Puella Magi if transformed. This is particularly why Kyuubey is so invested in getting Madoka to make a contract, because her potential is the greatest out of the main characters. Puella Magi who contract with Kyuubey gain a weapon and a special power, the latter which corresponds to their wish. Sayaka Miki, for example, wished to heal the hand of a boy who could no longer play the violin. Her special power was fast-healing regeneration, as her wish was to heal someone. When Puella Magi fight Witches and successfully defeat them, the Witch drops a Grief Seed upon it's death. The Grief Seed is the only thing a Witch leaves behind when they are defeated and are used to replenish the Soul Gem of a Puella Magi. Kyuubey says that when a Soul Gem is corrupted, through a Puella Magi using too much magic or feeling immense despair to the point of no return, a Puella Magi will die. The truth is far more sinister, but before getting to that the Soul Gem should be explained a bit more in detail.
Puella Magi; The Soul Gem
A comparison between a Grief Seed and Mami Tomoe's Soul Gem.
The Soul Gem is just what it says on the tin can - a gem that stores a soul. When Kyuubey makes a contract with a girl and makes them a Puella Magi, part of his job is to rip their soul out of their body and put it into this gem. The Soul Gem is what makes a Puella Magi capable of transforming and fighting off witches when they do so. According to the episode where this is revealed, Kyuubey states that the flesh body of a Puella Magi is more or less hardware. Their bodies can be damaged to extreme measures, but can be healed with magic. They don't even suffer much, if not any, pain because their Soul Gems absorb it. So long as their Soul Gem is safe they are pretty much immortal, which is why Kyuubey says that they are so qualified to fight against Witches. A normal human body would not be able to take all the damage a Witch inflicts when battling someone.
Of course that means that the Soul Gem is the most important thing to keep safe, which is why it attaches to a Puella Magi's body when they transform and is easy to travel with. Since a Puella Magi always keeps their Soul Gem on hand in case they need to fight Witches, they never learn that Kyuubey ripped out their soul without telling them about it. Their Soul Gems must normally be within 100 meters of their body, in order for them to function. Without it the body becomes nothing more than a corpse.
The Food Chain.
Witches spread misery and kill Humans.
Humans become Puella Magi to fight Witches.
Puella Magi ultimately become Witches.
What is missing in this chain is the last part. What happens when a Puella Magi finally has their Soul Gem corrupted. Unlike what Kyuubey said, the truth is that Puella Magi don't just die. Instead they become the very Witches they fight and the power of the Witch is dependent on how powerful the Puella Magi it came from is. Witches are not actually the girl, but their tainted soul. All Puella Magi are destined to eventually turn into one, when they finally fall to despair themselves for whatever reason. The only way to combat this is to fight the current Witches and harvest the Grief Seeds. Puella Magi do not actually know of this, as Kyuubey does not inform them of their fate, and believe they simply die.
Now that the basic principles of a Puella Magi have been covered, lets move onto Homura specifically.
Homura Akemi; Unique Powers
Homura's Puella Magi outfit. The diamond on her hand is her Soul Gem.
Homura is a rather unique case in terms of Puella Magi. Kyuubey himself says that she is an irregular of sorts. She is considerably the most powerful Puella Magi present, yet her powers are unknown. For most of the girls it's fairly obvious: Mami fights with matchlock guns, Sayaka with swords and Kyouko with a spear. Yet Homura never actually reveals what her weapon is. The reason for this - she doesn't have any weapons. Her 'weapon' is actually a shield that is attached to her left arm (not pictured), which she can use to defend against other magical based attacks. She also has a very weak purple blast power, based on what we see in episode 1 and 10, but doesn't use it often at all because of just how weak it is.
Yet there are some things about Homura's power that is very interesting. For one thing she can seemingly teleport from one place to another, requiring little effort on her part. In episode 3, when she is fighting Charlotte' second form, she repeatedly dodges the Witch's attempt to swallow her whole by appearing from the top of one large needle to the next. In episode 5 and 6 she is also seen teleporting from place to place when dodging Kyouko's attacks and stopping the latter's fight with Sayaka. However there is an interesting aspect with episode 5 that might be overlooked.
In this image we see Homura movements reflected by the water droplets, as she rushes to stop Kyouko and Sayaka from fighting. Water, as one knows, is very fast when in motion. Yet Homura's speed even defies their own. From this it can be inferred that Homura is not actually teleporting, but moving at incredibly quick speeds that cannot be perceived by the normal human eye, and at a speed that is even faster than a falling water droplet (the drop is still moving itself, but not nearly as quick as Homura).
So Homura is somehow moving at a very fast speed, but is not being noticed by anyone around her when she does this. She doesn't even make a sound when she moves from place to place, nor does she leave any kind of tailwind, so somehow she is going from one place to another without anyone even realizing it. Then, in episode 8, we get the answer from Kyuubey: time control magic. Specifically, the power to stop time.
Through a mechanism in her shield, Homura is capable of stopping time around her for a short period. She can also bring people into this world of frozen time if she is in physical contact with them when she activates her shield. During this time she can still use her powers and weapons, though the actual impact of the weapon only occurs after Homura unfreezes time. So if she fires a bullet at someone, it will only travel as far as a few inches in front of them until she unfreezes time. Once she unfreezes it the bullet will hit the target, unless they have some kind of amazing luck or power to avoid a bullet inches from their face. She can also physically damaged someone or something, but the actual feeling of being injured will only occur for the other person after she unfreezes time.
Homura Akemi; Time Traveler.
tl;dr: Homura is a fourteen year old female Doctor Who.
Her time control magic also allows her to travel backwards in time, into a past timeline. She is not actually rewriting time, but is travelling to a parallel universe and living within that universe. The only limitation on this is that she can only go back as far as a month in time. This is, for obvious reasons, capped off while she is in Luceti. She can still stop time for as long as a minute (which is about the maximum time she has in the series).
Before the main series began, Homura arrived to Mitakihara Middle School as a new transfer student. She had heart problems for many years, but was finally discharged from the hospital and allowed to attend school again. Her lack of social communication during her time as a patient made her very socially awkward and unable of performing tasks that required rigorous amounts of physical exercise (attending P.E. class for example), which gave a fairly low self-esteem. After her first day of school, as she walked home, she was targeted by a Witch and was almost killed. Thankfully she was saved by her classmate, a Puella Magi named Madoka Kaname.
In the previous timelines before the series began, Madoka contracted with Kyuubey before Homura met her and became a Puella Magi alongside Mami Tomoe. Homura became friends with Madoka and looked for ways to repay her friend for the great deed she did in rescuing her. That way came when, after that timeline's Walpurgis Night, Madoka died one month later and Kyuubey came offering Homura a contract. Homura accepted and wished to redo her meeting with Madoka. As the special power of a Puella Magi correspond with their wish, Homura's power was time control. Using the power from her wish she traveled back one month in time, to the day she was discharged from the hospital.
She went through four more timelines afterward, which were all documented in episode 10. The results were pretty much the same - Homura comes to the middle school and meets Madoka. Both proceed to fight Witches, alongside Mami, Sayaka and Kyouko. When Walpurgis Night comes everyone dies except for Madoka and Homura. Madoka dies/becomes a witch and Homura time travels back. It is only in the 3rd parallel timeline Homura goes through does Madoka realize Homura is a time traveler. Nearing the end of her life, she asks Homura to stop her (Madoka) from contracting with Kyuubey. Homura promises to help Madoka, no matter how many times she must travel through time, and then shoots Madoka point blank to prevent her from becoming a witch. It is after that does Homura's determination rise and she turns cold, in order to accomplish her goal of saving Madoka.
Resources/Other Links
❖ Puella Magi Wikia Page (a very good source of information)
❖ Madoka Magica Official Site (in Japanese)
❖ Madoka Magica Manga (each chapter covers one episode of the anime. good for a quick dose of the series)
❖ Madoka Magica Episodes (all on stream)