There are some interesting references to paper, origami, cards, etc.
in House of Cards.
Perhaps for Claire, it's a reference to two things - one being her suppressed creativity and how that ties in to her longing for Adam.
We don't know that much about her past, except that which has been revealed through her conversations with Adam.
We also learn a little about her choices when she sat by Steve's bedside as she discusses the conscious decision she made to be with Frank, not necessarily for love, but for the respect and freedom he would give her.
Ironically, though Claire is not locked into a life of being the trophy wife, she is seemingly trapped in a marriage of partnership and reason.
Adam (and the origami) represent freedom, art, and impermanence.
All that which she does not possess.
The second reference may be to the constant folding and unfolding of events in her life.
As close as she may be to Frank, he is still able to use her as a pawn to further his own goals.
She makes decisive efforts to curtail this when she sabotages the water bill.
Paper is temporary; it does not provide real shelter, as was mentioned by one of the workers at the shipyard meeting Peter Russo held.
Nor does it protect us from the elements.
It can all collapse in a moment, just as a structure made of cards will do.
Money, made of paper, can be easily burned, destroyed or lost.
His kids also do origami - a foreshadowing of the two tiny victims who will pay a huge price for Frank's lust for power.
The homeless guy took a bill - a physical representation of what our society deems to be worth something, folded it, and turned it into a piece of art for Claire.
It wasn't a small amount for a homeless person - it was $20!
A blatant reminder that those things which are valuable to some, may mean nothing to others.
It was an eerie reminder that her life and everything she values can quickly fall apart.
Lastly, Frank in his speech when he was honored at his alma mater talks about harmony, and how "it is not about what's lasting or permanent,
it's about individual voices coming together for a moment.
" This is so contradictory to everything Frank truly stands for.
The reality is, he backstabs, manipulates, and deceives everyone around him, especially those who are near him.
The only one he does not deceive is Stamper (which should be discussed on a separate thread as that is an interesting theme as well) Frank is fully driven by the hunger to establish legacy, permanence, and power - basically the opposite of what origami represents.
I would love to hear more feedback from others
- I think this is a really interesting concept and there is so much room to dig deeper.
movies.stackexchange.com/users/4169/Lisa
Another aspect to origami is that it requires precise, painstaking execution, which seems to come naturally to Claire.
Like origami, she is beautiful, complicated, and controlled; given the way the plot unfolds, she may be crushable.
Also, the folds and doubling-back suggests to me complexity and double-dealing.
I find her very intriguing.
Although I am associating the origami with negative connotations, I started liking her when she took up the origami, and found myself falling under her sway.
DRJ
movies.stackexchange.com/users/4257/user4257
It's a stork.
I think it foreshadows her interest in having kids.
Note that she exchanges it with Russo's kids, whom she dreams about just before she shows overt interest in pregnancy.
movies.stackexchange.com/users/4435/joel
Coming late to HoC party, but very intrigued by the paper references.
I think it can also extend into the newspapers as well.
Their impermanence and the digital forms of information that now are of so much weight and importance.
The significance of the origami is far reaching.
It represents a childlike whimsicalness, referring to Peter's children and to Claire's missed opportunity at motherhood.
It is the transformation of a flat piece of ordinary paper into a beautiful unique hand made 3 dimensional art object.
With the same generic piece of paper, you can imagine anything and create it from almost nothing.
It is a swan that the homeless man creates for Claire.
Not a crane.
Swans are often folded for weddings.
The folds allow for patterns on a piece of paper to be incorporated as the illusion of feathers ( as is shown in the folds of the US $20 bill).
Swans mate for life.
Swans also signify a journey or that a is change coming.
For better and worse Cliare is married to Frank for life.
She is on a journey with him and change is always afoot.
When Claire leaves Adam in New York, she rearranges the photograph he has printed for her.
Again the impermanence of paper, who prints photos anymore and who prints them that large?
More paper falling by the wayside.
She also folds a swan and leaves this for Adam showing her allegiance to Frank.
movies.stackexchange.com/users/8328/user8328
I think it may also be a representation of Claire herself.
Paper on it's own is blank and undefined but once folded and manipulated it can serve a purpose as with the origami.
After the photo Claire takes of the girl is first printed Adam keeps projecting what he sees onto the image, wanting to name her Claire indicating that he sees her this way.
Claire before leaving however, rearranges the photo into how she sees herself.
A tightly controlled image surrounded by obscure and indiscernible border.
movies.stackexchange.com/users/8029/A V
I do believe it to be an allusion to the meeting of Diogenes of Sinope and Alexander the Great which is one of the most well-discussed anecdotes from philosophical history.
Many versions of it exist.
The most popular relate it as evidence of Diogenes' disregard for honor, wealth, and respect.
Alexander says to Diogenes "tell me what you want and I will grant it".
Diogenes looks at the great man, who is standing right in front of him. "
Grant me anything?
Then move to one side as you are blocking my sun!"
movies.stackexchange.com/users/41017/Scott
In the 6th episode of House of Cards, we see Claire have some sympathy towards a vagrant outside of her work and gives this man $20 to get some food from a deli close by    The next day, the man is still there and then flicks back the $20, now in the shape of an origami crane.   What is the purpose of this scene, and what does the Crane represent given the current events in Claire's life?
What is the significance of the Origami Bum?
movies.stackexchange.com
<analysis><house-of-cards>
movies.stackexchange.com/questions/10050
movies.stackexchange.com/users/130/Tablemaker
It may also be a representation of Claire herself. It may also be a representation of Claire herself. 
It may also be a representation of Claire herself. Claire takes a photo of a girl and Adam imagines her image as a representation of her. After the photo is first printed Adam keeps projecting what he sees onto the image. Claire rearranges the photo into how she sees herself before leaving.