To make the chalkboard write on itself, the special effects team put a magnet in the chalk and put a magnet on the other side of the chalkboard to make the chalk move--it was very difficult to get a small magnet inside the chalk and make it write smoothly.
The producer said that it worked out well that Gabriel and Alex had worked together, and Alex and Milly knew each other from school, because it made Toni the outsider, which mirrored Annie's character and feelings of alienation within her own family.
In Peter's first scene at school, the words "Escaping Fate" is on the chalkboard with the teacher discussing it. This is a reference to Halloween (1978), where the main character discusses the same thing in class. Appropriately, this movie was released the same day as the trailer for Halloween (2018).
The house was constructed completely on sets on a soundstage in Utah in order to follow Aster's shot list. They needed to be able to remove walls and ceilings in order to shoot the rooms to look exactly like the miniatures.
Production designer wanted to play with the idea of "sacred geometry"--triangle (Annie, Peter, Charlie), square (introduces Steve's character, home, groundedness), circle (infinite, genesis, Ellen) - all different shapes embedded within the design of the set. If you look closely, the second floor hallway has squares and triangles carved into it.
The trivia items below may give away important plot points.
In an interview, Alex Wolff explains that he wanted to actually break his own nose for the scene where his character slams his head into a desk. Director Ari Aster respectfully declined that offer and told Wolff they'd give him a soft, cushioned desk for the scene. When it was time for the scene to be shot, Wolff slams his head into the desk only to discover that the top was foam and the bottom was hard. He dislocated his jaw (which is a previous injury the actor has had) for the scene.
During the support meeting, Annie recounts her brother's suicide at age 16, and states that his suicide note blamed their mother (Ellen) for "putting people inside him." Though Annie chalks this up to his schizophrenia, it could very well be that Ellen originally attempted to conjure Paimon through her own son. His death (and her failure to summon Paimon) would then explain why Ellen put so much pressure on Annie to have children, and why Charlie stated early on in the film that her grandmother wished she was a boy.
Throughout the film, several words can be seen scrawled on walls. At one point, there are two words, LIFTOACH PANDEMONIUM. Liftoach is an English transliteration of the Hebrew word 'To open', Pandemonium is Latin for All Demons and is what Satan names Hell in Milton's Paradise Lost.
During the party scene, just as Peter enters the bedroom to smoke pot, some kids are watching a black and white video on a laptop of someone being beheaded on a guillotine - foreshadowing the film's multiple beheadings.
The script was originally written to take place in a snowy landscape in the mountains, but due to cast availability they had to shoot in May/June, so they decided to "embrace the green" and look at the lush, green, springtime as emblematic of the awakening and rebirth of the cult.
At the end when the life-sized figurine of the demon Paimon is shown, the fingers of its right hand are positioned in a way that Jesus is often portrayed as doing in medieval paintings (pointer finger and middle finger out and together, other two fingers curled in a fist, and thumb parallel but slightly curved), but on Paimon, he's holding it upside down. The medieval and ancient hand gesture is used as a representation of Jesus, so like crosses being inverted, Paimon is using this to disrespect Jesus. Also, Paimon is wearing a halo with rays coming out of it like a sun. This halo is also seen in depictions of Jesus, thus meaning those worshiping Paimon believe he is the true savior.
Annie's mother Ellen likely tried to summon Paimon through her son Charles before his suicide, which is why he claimed she was trying to put people inside of him and was misdiagnosed with schizophrenia. Charlie was the reincarnation of Paimon, and shared a similar name with her uncle.
During the therapy session Annie tells of a troubled family history that she had largely kept to herself, because she did not want to "put more stress on the family". It is suggested that the family had been under the influence of the cult for a long time and hence that Annie was the one suffering for the longest time. During the films finale, Peter gets to see with his own eyes what had been manipulating his mother all those years. The song playing over the end credits, "Both sides now", thematically relates to this series of events. The song contains the line "I've looked at life from both sides now" which could suggest, that Peter was now able to see things from his mother's point of view. The song also contains the line "Tears and fears and feeling proud, to say I love you right out loud" which could suggest, that Peter finally understood that his mother actually did care for him and now was able to return those feelings, in spite of all he had been put through.
In an early scene, the subject of free will is discussed while mentioning one of Sophocles' plays. In the same vein, Ari Aster states that the Graham Family has no free will on the events that are taking place. For Aster, the film is very Greek in that sense and the way things turn out is absolutely inevitable, the family has absolutely no agency. And that's where the dollhouses came in. Annie creates these miniature figures and dollhouses and they served as a perfect metaphor for the situation; they're dolls in a dollhouse being manipulated by outside forces. Any control they try to seize is hopeless.
According to Ari Aster, with Hereditary, he wanted to make a film about the corrosive effects of trauma on a family unit and he also knew that he wanted to make a film that had sort of an ouroboros quality about a family that's basically eating itself in its grief. Ultimately, Hereditary is a film that was seriously tackling these issues and operating almost as a meditation on these things, while at the same time functioning as an exciting genre film that hopefully delivers.
In one of the scenes in Peter's classroom, the teacher is talking about the sacrifice of Iphigenia, Agamemnon's daughter. Both Iphigenia and Charlie are daughters who were sacrificed in hopes of bringing forth a great event; one being victory against the Trojans and the other the bringing forth of Paimon.
For Ari Aster, the film itself is a running metaphor for family trauma and grief. It is operating all the way through and at the end, the movie is still about how trauma can utterly transform a person, and not necessarily for the better.