Bob, the short answer is: yes, do train the same muscle every 24 hours if it pleases you. I think you would see great improvements.
The long answer is that this is quite a subjective topic, and your goals are a consideration here. Let's discuss why.
Some of the strongest men in the world (olympic weightlifters) often train the same muscle for hours per day, seven days per week. Their training volumes are greater than you or I can likely imagine.
On the other hand, the biggest men in the world (bodybuilders) are concerned with 48 hour recovery windows because muscle damage, soreness, perceived exertion, strength deficit, and so on often peak within that time frame (according to studies -- and anecdotes of course). It's a concern for bodybuilders because they wish to have a "full tank of gas" when performing their sets in order to gain maximal muscle size. These gentlemen/women eat 8,000-10,000 calories per day and wake up multiple times during the night to ingest more calories and protein. Again though, these athletes have training volumes greater than we can imagine. The average Joe Gym Rat is leagues away from concerning themselves with this.
This is a complex question for many reasons, and that's why I say it's subjective. For example: how much exertion is actually required to warrant 48 hours of recovery -- just one set to failure or 40 sets to failure? How does this window vary between well-trained individuals and untrained ones? We know hormone levels have an adaptation period as we get in shape, so how big of a part is our current hormone level playing in recovery? Are we an individual that has naturally high testosterone levels? How much sleep have we had? How much protein/amino acids? How much training volume did we do last time? The body is quite complex and we're basically just trying predict the weather here. We shouldn't be talking in absolutes, or arbitrary numbers. It's hard to say how all the systems are going to work together in your case without study.
All of that said, the easiest thing to do here is just to try it out and measure your progress. You may start to plateau while training under this modality, in which case you would incorporate longer rest periods. On the other hand, your physique might start looking awesome, but your lifts may not improve much. Some things to measure are: circumferences of muscles and waist, body fat %, 1RM numbers, etc. In any case, I think you would be well into the advanced/elite athlete category before you should even begin thinking about this. In my experience, more training volume precipitates better results. After all, elite athletes train all day, every day.
Just a side note: be wary about injury. Training at higher volumes can be dangerous, especially if you are not using your body for most of the day. Muscle tissue is not the only thing we are concerned about here. Be sure you are adequately prepared for your lifts by building up to them.
Here is some further reading on the 48-hour window: