Summary

Clock in matchbox size with touch control

Binary clock, it has 16 pieces WS2812B matrix display showing binary coded time.

Touching the top, the Arduino gets a wake up interrupt from a TTP223 touch sensor, wakes from sleep, reads the time keeping chip PCF8563 and displays the time on WS2812 chain. Waits 2 sec, turns off the WS2812 supply and goes back to sleep mode to save the battery. In this mode, the consumption is about 30 uA, it means that the battery is good for 20000 working hours (many years theoretically) with a single charge.

The color of the WS2812 is freely adjustable in the Arduino program, the default is red for the 1, yellow for 2, green 4, blue 8 binary weight. To read the time, you add the weights vertically.

When the charger is connected, the display stays on. Holding your finger on top keeps the display on, gradually increasing the intensity to maximum.

Display: The matrix of 16 WS2812B LEDs in series is available on Aliexpress. Connect 3 wires, Gnd, Vcc, DI. The display has many 100nF capacitors built in, and this was a problem since switching the power on to the WS board resulted in a high current spike, the voltage fell, and the program crashed. The 470uF capacitor Gnd-Vcc solved this issue, so change the 10uF capacitor on the PCF8563 board to 470uF tantalum.

TTP223 : Make the A jumper (active low output). Remove the LED or its resistor. Use no pins, the space is limited.
Available here: https://www.aliexpress.com/store/910410007/search?SearchText=ttp223

PCF8563 clock module : Cut the module in half as shown, we don't need the battery since the LiIon will function as backup battery. Cutting the PCB breaks the VCC path, restore it with a jumper.
Use no pins, the space is limited.
The module has no pullup resistors, you may build in these, but the circuit seems to work without these, perhaps the PCF library turns the internal pullup resistors on.
Available here: https://www.aliexpress.com/store/910410007/search?SearchText=pcf8563

TP4056 : Remove/replace the 1k2 resistor with 6k8 to 10k. Solder 2 identical 100k resistors on the USB input + side to GND forming 1/2 voltage divider to Arduino A0. Order the smallest possible TP4056 module. Use no pins, the space is limited.
Available here: https://www.aliexpress.com/store/910410007/searchorigin=n&SortType=bestmatch_sort&SearchText=tp4056

Arduino Mini : Remove the power LED or its resistor. Remove the unused 5V LDO chip (LM1117) as well to avoid power leakage. Use no pins, the space is limited.

Construction : Print the 3D parts. Use 0.25mm enamel wire for wiring. The WS matrix is followed by the Mini board and the clock board in the case. There is a cavity for the TTP223 and the 6x6mm tactile switches. See picture.

The battery compartment is made for the Ipod 616-0232 battery in this design. You may use any other type of Li-Ion battery, 3D print a box for it using my wedge assembly or simply glue it to the clock case. Drill or burn a hole for the wires to the Arduino.