There is only the rhizome for this type of begonia (no corms or tubers). It takes time for a young plant to develop longer rhizomes, more rhizomes. You can also use a leaf (a mature leaf is better than a new leaf). You can divide this leaf into wedges by cutting off the petiole first then cutting between the main veins (like a slice of pie). Each wedge can be inserted into a good growing mix or perlite. Water in, cover with a lid or plastic wrap. Do not let it go totally dry nor keep it sopping wet.

Here is what a mature rhizome should look like (in this case it is Tangalooma). The rhizomes are getting so congested that it is a good idea to thin them out and use the cut pieces for new plants.

I would be more patient and let your plant grow for a year or so before trying to make more plants out of it using rhizomes. On the other hand use some of your more mature and "ugly" leaves for propagation instead.

Here is a small leaf rhizo that I just let it do its thing.

It's a big plant almost 2 years old.

Except that is Gryphon, not heracleifolia. You can easily cut the stems into multiple sections and start new plants with those.

You mean that I discard the top portion of the plant leaving main stem then cut the main stem into 3-4 longitudinal sections in a way that each section has some roots attached to it and then plant them separately? right?

You can cut the stem back fairly hard (say an inch or two above the soil leaving a few nodes to branch out). Cut the tops off the leggy stems but keep a couple of inches of the stem with the tops and use these for new starts (simply push the tops in a good potting mix or perlite). The leggy stems can then be cut into 4 inch lengths or so (have a few nodes on each cutting). Lay these horizontally on a bed of perlite or good potting mix. Keep the leaf side of the stems up so the bottom part will root. Water in and add a plastic lid or baggie to keep the humidity high. They should root in a couple of weeks. Do not keep soggy wet though.

Sorry but want to be sure that I don;t make any mistake. I don't want to loose this beautiful plant. cut the stem back fairly hard perhaps means I cut the whole plant leaving 2 inches of stem and roots in the soil intact and this 2 inches of stem should have few nodes. I didn't understand what do you mean by 'Cut the tops off the leggy stems but keep a couple of inches of the stem with the tops and use these for new starts'. By the way few months ago several new plants grew around the main plant which we gave to our friends, may be in future same things happen again. The plant we have right now is quite big and when I dig it out to look for tubers or bulbs, it is not standing upright and leaves have also become somewhat distorted ( bottomline is that it doesn't look as beutiful as it was before)and I want smaller plant which grows symmetrically and looks beautiful. thanks

This begonia is probably considered a thickstem, maybe a shrub but is not tuberous so there are no tubers associated with this type of begonia.

Yes, cut it down leaving some stem with nodes (where leaves once were). The stems should send out new shoots from the nodes. Also this should encourage new shoots from the roots.

As for the tops and stems, when you cut away the top portion you will have a long stem. You can plant it this way in one piece but I thought the idea was to shorten the entire plant. To do this you will cut the top with some of the stem (remember the nodes where roots will grow). Now you have the stump with roots, the top with a short stem, but you also have the stem left over from the two cuts. You can cut this into short sections (just make sure you have some nodes on each section - 2 to 3 nodes are minimal). If you don't remember the orientation (which end is down and which end is up), then you can lay these sections horizontally. Since this is not really a rhizomatous either then it really shouldn't matter if there is a side that will root and which side is leaves only. If you do know the orientation of the stem then you can insert the bottom end into your potting mix.

I've done this earlier this year where I cut one of my Gryphons down pretty hard. cut the tops out and used Jiffy pellets (expanded peat cubes) to push the tops into, and laid the middle sections on a bed of perlite on the ground - most have rooted and some have already put out new shoots.

I don't have any pictures of these but here is one of a manicata stem in a Jiffy pellet.