I don't know anything about sensing a heartbeat, but I assume that's no problem.
I know its really abstract, I'm new in electronics.
I'm an electronics guy, and to me this sounds like an electronics project (and a programming project, which is part of many electronics projects and part of every Arduino/microcontroller project). So, here's my suggestion: Partner-up with an engineering student. He/she can do the electronics & programming, and you can to the artistic part! I assume that's allowed for a BA thesis, and I assume you can both get credit.. I think it's pretty common for PhD candidates to get help from various specialists as needed. And if you each want to own one of these things, building two of something is only slightly more work than building one (although it's twice the parts & material cost).
An engineering partner could really help with all the stuff you are asking about here!
...without using a laptop in the circuit.
Why not? A computer could add a LOT. The Arduino is great at "sound controlled lights" (I've done 3 such projects), but it's not that great at generating/controlling sound. You can generate simple sounds, but it requires an external digital-to-analog converter for high-quality sounds, and it doesn't have much memory for sound files. With a computer, you could use MIDI to generate synthesized synchronized to the heartbeat... It could be "synthesizer sounding" techno-like music or realistic-sounding orchestral or jazz music, etc.
Then for lightning, I think, dmx would be the best option, because I don't want to use a 220v dimmer
In my projects I haven't used DMX, because it's an extra transmitter & receiver and extra programming for the DMX protocol. It's going to be more expensive than building something directly-connected* from scratch. The advantage of DMX is that you can buy DMX-controlled lights, or DMX controlled dimmers. And, you can even buy a DMX controller that plugs-into a computer. That greatly reduces the amount of electronics you have to build.
...and led's are too insufficient or too expensive.
Actually, LEDs are worth considering. One of my projects was a "giant VU meter" made out of regular 'ol "high brightness" LEDs. It's plenty bright enough for in indoor lighting effect. For light output, you can get "high power" LEDs (1 Watt or more). These are the kinds of LEDs used in home lighting and DJ lighting effects. High power LEDs require a special power supply, but you can buy it, and in some ways they are easier to dim/control than incandescent lamps and the 120-V/220V power is isolated from the low-voltage control, so that's wone less thing to deal with (engineering considerations). With RGB LEDs, you can have color-changing LEDs that generate "any color of the rainbow", and of course LEDs can flash on & off faster than incandescents (artistic considerations).
Maybe there is a way to control a party strobe light with arduino?
Yes. That can be done. For example, strobes used in photography are synchronized to the shutter. Or, I'm sure you can find one that's DMX controlled, or you you could build/hack something. Electrically, traditional xenon strobes require fairly high voltage (maybe 400V minimum), so if you build one from scratch, that's extra circuitry to boost the supply voltage (and extra danger) but there are also [u]LED strobes[/u]. (That one runs off 120V/220V power, but I'm sure the LEDs are lower voltage internally.)
- You can't "directly" connect the Arduino (or a computer) to 120 or 220VAC. You need an opto-isolator, or a solid-state relay (which is optically isolated).