Tortoise shells are of two kinds.
One of the bony or the cartilaginous kind and the other of the protein chitin.
The bony and cartilaginous kind is the one which you find on those reptiles which have been on planet Earth for millions of years and have this in-built talent to see at least four generations of a human rise and fall. These shells come in various different colors ranging from a drab brown to a bright yellow (especially in case of the Indian endemic forest dwelling Cane-turtle Vijayachelys silvatica). They vary in shapes with some being flattened and some rising like symmetric cones. They also carry astounding patterns on them like the peacock-shell turtle (Nilsonnia hurum) and the green turtle (Chelonia mydas), the patterns which are similar to the Tyndall effect caused by the sun when it is right above the horizon. These bony shells are the skin (outer most covering) and serve as the first line of defence for these reptiles.
But, the ones made of chitin are far more common, elusive, tiny, can open out and help the animal fly.
The tortoise shell beetles are the ones which posses these shells.
They are insects belonging to the subfamily Cassidinae. These insects are beetles not bugs. There is a major difference between the both of them. Most insects posses two pairs of wings- A forewing and a hindwing. In the case of bugs, both the pairs are membranous and their mouth-parts are usually modified to feed on liquid diet (like aphids, plant hoppers and stink bugs). In case of beetles, they are chewers, their mouth parts are modified to cut and chew and their fore-wings are hard (like shells) to protect the fragile hind-wings.
Some common beetles which you will see, if you live in India are the Rhinoceres beetle, the Jewel beetle and the water beetle.
If you live close to an area with a lot of trees, there is a very high chance of you encountering these beetles. They can be as small as a mustard seed or as big a soda bottle cap.
I encountered one right outside my home, it had the most amazing green sheen which I have seen.