For now, though, the London-bound athlete has settled for a silver necklace of the famous symbol, which she twirls excitedly in her fingers.
While Turkey excels at wrestling and weightlifting -- the nation's men and women have won 67 Olympic medals in the sports -- never before has a gymnast participated at the Games in Turkey's red and white national colours.
"All you really need is faith in yourself," the 22-year-old gymnast told Reuters in her dormitory room.
Uctas' Olympic achievement is all the more noteworthy given that as a nine-year-old -- a vital stage in a gymnast's development -- she survived a catastrophic earthquake which struck north-west Turkey in 1999.
That earthquake killed around 17,000 people and left hundreds of thousand others homeless, including her family.
"We lived in a tent for a year, in constant fear because of aftershocks," Uctas said. "But I never stopped training, even when the gym was being rebuild, I trained outdoors and remained focused on practising as much as I could."
Pinned up on Uctas' wall is a pencil drawing she made for motivation and inspiration, of her heroine Svetlana Khorkina, Russia's Olympic gold winner. Uctas has a talent for drawing, and loves sketching gymnasts, as a way of relaxing after her taxing physical training.
Turks have attended all the summer Olympics since 1908 bar three, but have won medals in only six sports.
"Turkey's policy in sports investment has been based on medals, and the budget for sport has always been distributed according to success," Professor Settar Kocak of Middle East Technical University, who teaches philosophy and history of sports, told Reuters.
"The money has always been invested in sports of most public interest, like wrestling, that Turks have been fond of for a thousand years," Kocak said.
Oil-wrestling, where the wrestlers cover themselves in olive oil and wrestle in open air on grass, is hugely popular in Turkey and some former Olympic wrestlers practice it after retiring.
An annual oil-wrestling festival in Turkey's northwestern province Edirne draw tens of thousands of spectators each year.
Turkey has bid to host the 2020 Olympics in Istanbul and is one of three cities in the running along with Tokyo and Madrid.
Certainly Uctas' qualification will help broaden the Games' appeal among her compatriots.
Uctas and her coach Mergul Guler were meticulous in their studying of the routines and the skills she would need to master to have a chance of competing.
"I went through so many regulations material on the internet to find out how to qualify for the Games. The reason why no Turkish gymnast ever went to the Olympics so far is not incompetence, it's just that no-one actually believed it could happen, and therefore never found out how to," Guler, a former gymnast herself, said in disbelief.
Uctas and her coach have modest hopes for now.
"The goal at the Olympics? Going there was the goal. But making it to the finals would be incredible," Guler said.