On July 16, 1945, the United States became the first country to successfully detonate an atomic weapon, signalling the beginning of a new era in warfare and in politics. This detonation took place in the middle of the New Mexico desert, with the bomb placed carefully atop a 100 foot tower. The bomb, nicknamed “Gadget”, had a yield equivalent to 20,000 tons of TNT. Just 24 days later, a functionally similar bomb (using Plutonium, unlike the Uranium bomb at Hiroshima) was dropped on Nagasaki.

No one was completely sure what would happen when Gadget went off. For a while, there was worry that the chain reaction would be unstoppable and react with the entire atmosphere. Before the test, Enrico Fermi took bets from some of the physicists and high ranking military personnel on whether the bomb would destroy the whole state of New Mexico, or the entire planet. The math seemed to show fairly conclusively that the world wouldn’t be destroyed, but a lot of the guards who didn’t know that became anxious. Kenneth Bainbridge, director of the Trinity Project, was not amused with Fermi scaring all the guards.

When the bomb was detonated, it left a crater of radioactive glass in the desert that was 10 feet deep and 1100 feet wide. About 240 people on the project directly watching the blast reported the early morning dawn being lit up brighter than full daylight for one to two seconds and felt a wave of heat roll over them that was “as hot as an oven”, even at a distance of 10 miles away. The shock wave took 40 seconds to propagate to the observers and was felt up to 100 miles away. The enormous mushroom cloud was 7.5 miles high. It was at this point that Bainbridge remarked to Oppenheimer, “Now we are all sons of bitches.” Oppenheimer later spoke his famous line, “Now I am become Death, the destroyer of worlds”, a quote from the Bhagavad Gita.

This famous picture was taken 16 milliseconds after the Trinity bomb exploded. That’s the hypocenter of the blast. Once the public found out about the bomb and where it was detonated (sometime during the late 40′s), they began traveling to the site and collecting the glass as souvenirs for themselves and to sell to tourists and collectors. This area was still lightly radioactive, and the government didn’t like the fact that people were carting off lots of the stuff or sniffing around their test sites. In 1953, the government bulldozed the site, burying any glass that was left and fenced off the area. A law was passed making it illegal to collect samples from the area. The only exception was that it was legal to buy and sell the glass that had already been collected and was already on the market. People began calling the collectible glass “Trinitite”.

The Artifact

You can still buy Trinitite today, on places like eBay or from mineral collectors. You can also buy it from United Nuclear, which is where I got my sample. United Nuclear makes two claims that I wanted to verify: that the sample was real and that the radiation level was safe. Apparently lots of people sell fake Trinitite, because it’s easy to fake and people will buy it. United Nuclear says that they check all the Trinitite they buy with a mass spectrometer to verify authenticity. Here’s what my sample looks like: As an undergrad at Georgia Tech, I have access to some unique resources. One of my friends happens to be studying Nuclear and Radiological Engineering, so I asked him if he could measure my sample with a Geiger counter to make sure I wouldn’t get cancer or to confirm that I would get super powers with this thing sitting in my room. He grinned and said, “Oh, we can do better than that.”

The Experiment

After talking to his professor, he got permission to run a bunch of tests on my sample for his lab class. It turns out that a Geiger counter wouldn’t be able to tell me very much about the sample, including how dangerous it really was. Instead, a high purity Germanium (HPGe) detector was used. HPGe detectors are large, expensive machines that must be cooled with liquid nitrogen. Here’s a cross section of a commercially available detector:

“Coldfinger” is also the name of the villain in the next James Bond movie.

Because the detector is so cold, electrons have a lower probability of escaping, which allows for a higher resolution and higher efficiency. The reason Germanium is used is because of its useful properties as a semiconductor. As incoming radiation hits the Germanium, it creates free electrons and holes. An electric field across the Germanium causes all of the electrons to get pushed to one side, creating a current. This current is proportional to the number of holes created, which is proportional to the energy of the incoming radiation. This current is read with a Multiple Channel Analyzer (MCA) that bins the energy from the detector into multiple channels. The efficiency of an MCA changes based on the energy range, so they must be calibrated with known radiation sources before use. To determine if my sample was really from the Trinity test, its activity can be compared to the activity of specific radionuclides measured from glass collected at several other test sites. Gamma spectroscopy can be used to produce a plot that’s a little easier to visualize. To perform the experiment, the MCA was set at 4096 channels with a live time of 120 seconds and calibrated. The sample was placed in the shield surrounding the detector and measurements were taken for 80,000 seconds (22 hours). The sample was then removed and a background measurement was taken for 80,000 seconds.

Results

Here are the results, along with data from other test sites (as of 2013): Note that percent error here is not standard deviation, it’s how much the measured sample deviates from a known sample. Here is the measured spectrum: Analysis

The interesting peaks here are Cs-137 at 661.5 keV, Am-241 at 59.4 keV, and K-40 at 1400 keV. The specific activity of Co-60 is very low, which is the biggest clue that the glass was collected from Trinity, rather than another location. For example, at Reggane, Co-60 is 207.2301 Bq/kg as of 2013, while Trinity is closer to 7.771128 Bq/kg. The activity we see here is 1.0 ± 1.0 Bq/kg, which suggests that my sample from United Nuclear is real and came from Trinity! The other thing this data lets us do is see how dangerous the radiation from my sample is. To do this, we simply sum up the counts collected at every energy and divide by the amount of time the sample was measured to get an energy indiscriminate counts per minute. It turns out my sample has a total gamma activity of 1183.29 CPM ± 5.43 CPM. So how do we know if that’s dangerous? That’s a little difficult to answer, because CPM is relative. The type and energy of the radiation matters enormously. This analysis only looked at gamma radiation, since that’s the kind of radiation that is extremely penetrating and difficult to shield. However, gamma radiation is not as damaging as other forms of radiation of the same energy, like neutron radiation. My sample also emits alpha and beta particles, but they are very short range and are easy to block. Alpha radiation can’t penetrate your clothes and beta radiation can be stopped with only a few millimeters of aluminum. So, ignoring alpha and beta radiation, the damage gamma radiation does increases with energy, and we have a wide range of energies in this sample. If we assume all the gamma rays are 661.5 keV from the Cs-137 and you are 1 cm from the source:

The attenuation of photons in water (which closely approximates human tissue) is:  Multiplying those two numbers gives 5.49*10^-13 Gray/second, or 1.73*10^-5 Gray/year. The conversion factor from Grays to Sieverts is 1:1 for photons. 100 Rem is a Sievert. This means that if you keep this sample 1 cm away from you for the rest of your life you will receive 1.73 milliRem per year.

If you’re lonely, you can eat 200 bananas to get the same (radiation) effects as sleeping next to someone.

Bonus Points

There’s one more really neat piece of information we can pry out of the data. A naturally occurring isotope in soil is Eu-151. When Eu-151 captures a neutron from, say, a nuclear explosion, it turns into Eu-152. That’s a great way to estimate the neutron fluence (which is neutron flux integrated over time). So, if we can estimate the neutron fluence of my sample, and since we know how neutrons are emitted by the explosion at different distances, we can estimate how far away from ground zero my sample came from.

The math gets a little hairy here, so get ready.

Eu-151 undergoes an (n,γ) to make Eu-152 (it captures a neutron). The specific activity of Eu-152 is given by this equation:

Where φ_th is the thermal neutron fluence rate (n cm^-2 s^-1), a is the isotopic abundance of Eu-151, N_A is Avogadro’s constant, M is the atomic mass of europium, c is the Eu concentration in the soil, σ_th is the thermal microscopic cross section of the (n,γ) reaction, CR is the cadmium ratio for a nuclear reactor, and T_1/2 is the half-life of Eu-152. From [3], σ_th = 5300 barns, CR = 43, and c = 1.2 ppm. The cadmium ratio approximately accounts for the contribution of fast neutrons to the production of Eu-152.

After calculating the flux, one can solve for the distance from ground zero via this equation:

The specific activity of the Eu-152 was calculated by dividing the net counts by the duration of time over which it was counted, the mass of the sample, the efficiency for the energy of interest, and the branching ratio for the decay mode of interest.

After decay correcting the specific activity to the time of the explosion (68 years) and solving for the flux:

Solving for the distance from the center of the explosion:

Solving for the distance from ground zero:

So my sample was collected about 76 meters from ground zero. That means it came from somewhere along this circumference:

Future Work

Some kinds of Trinitite formed from sand on the ground turning to glass, and other kinds formed from sand being drawn up into the explosion, turning to glass, and then raining back down. If my sample was formed on the ground, we would expect the distribution of radionuclides would have an approximately continuous gradient. If it formed in the air, there would be a discontinuity in the distribution. This could be estimated with beta spectroscopy on each side of a sample. If the activity is spatially uniform, then the radionuclides are uniformly distributed. For this experiment, only gamma spectroscopy on one side was performed, so it’s unclear how my sample formed. However, a lot of the pieces in my sample appear large and flat, like they flaked off of the surface of the desert, so I would hypothesize that it formed on the ground. I would expect them to be rounder if they formed in the air.

Conclusion

It’s interesting to think of all the time and effort and money that went in to making the device that created Trinitite. It’s like I have a little vial of second order creations by Fermi and Feynman and Oppenheimer. I learned a lot about radiation physics and now I have a nice test radiation source for other projects that is pretty well characterized.

References

None of this would be possible without Nicholas Piper, who collected the data, performed analysis, and answered my questions. Also, thank you to his lab partners, Catherine Bartgis, Akshat Bhatnagar, and Benjamin Bane.

1. Tsoulfanidis, Nicholas, and Sheldon Landsberger. Measurement and Detection of Radiation. 3rd ed. Boca Raton, FL: Taylor and Francis Group, 2011. 384-385. Print.

2. Belloni F, Himbert J, Marzocchi O, Romanello V. “Investigating incorporation and distribution of radionuclides in trinitite.” May 5, 2011. Journal of Environmental Radioactivity, 2011.

3. Parekh P, Semkow T, Torres M, Haines D, Cooper J, Rosenberg P, Kitto M. “Radioactivity in Trinitite six decades later.” Wadsworth Center. University at Albany. Journal of Environmental Radioactivity, 2005.

Other interesting Trinity videos

The Planck Project recently published the most complete view of the cosmic microwave background yet. I thought it would be cool to show it with three.js on a sphere, so I used ImageMagick to transform the 2D Mollweide projection into a square image to use as a texture. Here’s the version I made. I noticed that my image transform still had one hole and some distortion near the poles. I contacted the people at the Planck Project who confirmed that the original image was a Mollweide projection, so I’m not sure why that distortion is there. About 24 hours after I hacked that together, I found thecmb.org, which did a MUCH better job than I did. I recommend you check it out, because it’s really neat. He even lets you scroll through the different spectral captures. His sphere also has a blurry patch that looks like it’s hiding the same kind of problem I was having. This is the page NASA has on how the data should be projected. If anyone knows what’s happening here, let me know. Here’s another neat page I found that shows 2D maps and lets you fade through different frequencies. So cool!

I built a little web app over Spring Break this year called Wait or Walk. The bus stops at Georgia Tech tell you how long it is until the next bus, but I realized one day that you don’t really care about that information. What you want to know is if it’s quicker to wait or just to walk. So I wrote a tool to answer that question. Wait or Walk calculates the time until the next bus and adds the time it takes the bus to get to your destination, plus time for stops, and then compares that to the amount of time it would take to walk. It uses NextBus to get bus times and an offline Google distance matrix call to get walking and driving times. It will also see if it’s cold or rainy outside and tell you the wait time if you want to stay warm/dry. It only works for Georgia Tech right now, but the idea can be pretty easily extended to other public transit systems, especially ones that are tracked on NextBus. The backend is written in Python using Flask and the frontend is all just javascript.

The Invention Studio at Georgia Tech is a fantastic resource that I love using. Tech recently filmed a short video about the Studio which you can see here:

They’re trying to get the word out so they can expand even further and keep inspiring students to make things.

I made a video using Gource and ffmpeg of the changes of the Robojackets Robocup git repository. It starts back in 2008 and ends with changes committed this month. I had to speed up the original capture because it lasted almost an hour. The original is on YouTube and linked at the beginning in case you want to look at certain periods a little closer. This is best watched at 1080p in fullscreen so you can see what’s going on. What it shows is the directory tree of our repository and each dot color represents a different kind of file.

The Romans were undoubtedly master engineers. They were experts at civil engineering, building roads, improving sanitation, inventing Roman concrete, and constructing aqueducts that adhere to tolerances impressive even by today’s standards. Perhaps the best evidence of their aptitude is the fact that many of those structures still stand today, almost 2000 years later. They even began dabbling in technology vastly ahead of their time. Hero of Alexandria drew up plans for a rudimentary steam engine in his Spiritalia seu Pneumatica. He called it the aeolipile.

It didn’t work very well. However, by the late 3rd century AD, all essential parts for constructing a steam engine were known to Roman engineers: Hero’s steam power, the crank and connecting rod mechanism (in the Hierapolis sawmill), the cylinder and piston (in metal force pumps), non-return valves (in water pumps) and gearing (in water mills). That got me thinking: Could the Romans have built a digital computer using only the technology and manufacturing processes available to them?

Maybe the first thing you would think of is a mechanical computer, like the Babbage Difference Engine:

While it’s a beautiful piece of engineering, it’s actually not a computer. It’s a calculator. Charles Babbage did design a mechanical computer, called the analytical engine. It’s never been built, because it would take up an entire room and be extremely expensive. I don’t think the Romans could have built the analytical engine or other purely mechanical computer because of the tolerances required. I don’t know much about their manufacturing abilities, but I know they didn’t do a lot of it and imported most things. I couldn’t find the tolerances needed for the Harvard Mark I, an electromechanical computer, but I’m hesitant to believe that they could have built that either. It’s hard to know how precise manufacturing techniques were back then, but one of the best clues we have is the Antikythera mechanism. It uses hand cut gears that are surprisingly precise, but still probably not good enough for a mechanical computer. Small inaccuracies in the gear trains would add up, and this is evident in the Antikythera mechanism. It would be even more pronounced in a room sized contraption and would almost certainly prevent any useful calculations from being performed.

If the Romans couldn’t do it mechanically, they would have needed a semiconductor. When most people think of semiconductors, they think of clean rooms and millions of dollars of machinery. However, that kind of equipment is only needed for high performance semiconductors used in modern integrated circuits with high speed, high efficiency switching. It turns out that naturally occurring semiconductors are actually quite common. Minerals like zincite, bornite, and carborundum will work. However, the best mineral to use is lead sulfide, AKA galena. It be used without any modification directly after mining. It’s got a band gap of about 0.4 eV. Ancient societies knew all about galena, and the Egyptians used it as makeup.

They could have easily made a cat’s whisker diode by using a small piece of springy wire to touch a point on the galena crystal, creating a crude and unstable metal-semiconductor junction called a Schottky barrier diode. Current will flow from the metal into the galena, but not the other way around. This is the same technique that was used in early crystal radio receivers. POWs in World War II used the oxidation on razer blades as a semiconductor and a safety pin to create a diode so they could build receivers to keep up to date with news on the war. It took a lot of patience to find a perfect spot on the blade that would work, and likewise it would take time to poke the galena with the wire in different spots to find a place that would act as a diode.

If they could have built a diode, could they have built a transistor? The first transistor that Bell Labs built (although not the first transistor ever built) was point contact and looked pretty similar to the cat’s whisker diode.

Instead of a single point, they had two, each formed by the contact of the edge of a piece of gold foil onto a hunk of Germanium. A simpler version of this can be done by basically building a cat’s whisker diode but using two springy wires touching two different spots on the galena. So couldn’t the Romans have just modified their diodes to make transistors? In my opinion, no. When Bell Labs was experimenting with transistors, they tried making a galena cat’s whisker version and ran into some problems. To being “transisting”, the whisker tips had to be closer than 0.1 mm. They also found that you had to use freshly cleaved galena surfaces and any humidity would interfere. They had to make extremely sharp whiskers by dissolving the ends with electrolysis. Now, I don’t want to underestimate the Romans. Maybe they could have trained workers to be that precise or come up with some other way of solving the problem. But we’re going to take it easy on them because there’s a better way to get around the problem of no transistors.

The Romans knew how to make wire and also knew how to work iron. So now the question is, can you build a computer using only diodes, wire, and iron? Well, prior to 1953, no computers used transistors. The Romans knew how to work glass, but it’s unlikely they could have manufactured vacuum tubes. While a lot of early computers did use vacuum tubes, they often also relied on diode logic. There are two major problems you have to solve with diode logic. First, diodes have a voltage drop across them, which means you need to amplify the signal every so often. Early designers solved this problem using transistors as amplifiers. The second problem is that you can’t design a NOT gate (an inverter) using only diodes. Designers got around this problem also by using transistors (or vacuum tubes). So how could the Romans have done it without transistors? Well, you can build an inverter using a transformer by simply flipping the secondary output wires around. A transformer is just a square iron ring with wire wrapped around each side. You have to use discrete pulses rather than continuous logic levels, but that’s how everyone did it in the 40′s and 50′s. To solve the amplification problem, you can use a relay. You can make a relay using only iron and wire, but they’re often small, intricate devices and they have moving parts. I think if they recruited Roman jewelry makers and scaled the size so it was reasonable to work with, they could have produced relays. There are some really impressive pieces of Roman jewelry that have been found. The image is too large to embed here, but take a look at the chain on this piece.

They also would have needed to make memory, a way to preserve the state of the machine. The obvious candidate here is core memory. Most core memory was made using ferrite, but regular iron can be used.  I won’t go into detail here on how core memory works because Wikipedia has a good article on it. If you’re not familiar, I recommend checking it out. It’s a really neat idea. Incidentally, you can also make logic using ferrite cores, like the Elliot 803 did, so that could be useful for supplementing (or even replacing) diode logic.

The last, and perhaps most important thing you need for a computer is electricity! We know that the Baghdad Battery existed back then, but it’s highly unlikely that a plausibly large array of them could power this hypothetical Roman computer. Instead, they would have had to use a generator. This is probably the most difficult part of this hypothetical computer. To turn a water wheel into a generator, they could have used a configuration like this:

But you need a magnet for that. Basically the only magnet they would have had access to is Lodestone. There are a couple of ways they could have made a better magnet. From Wikipedia:

• Heating the [iron] above its Curie temperature, allowing it to cool in a magnetic field and hammering it as it cools. This is the most effective method and is similar to the industrial processes used to create permanent magnets.
• Placing the item in an external magnetic field will result in the item retaining some of the magnetism on removal. Vibration has been shown to increase the effect. Ferrous materials aligned with the Earth’s magnetic field that are subject to vibration (e.g., frame of a conveyor) have been shown to acquire significant residual magnetism.
• Stroking: An existing magnet is moved from one end of the item to the other repeatedly in the same direction.

By iterating the process several times to make successively stronger magnets, the Romans could probably have made some magnets good enough for a generator.

It’s important to note that I’m not a historian, I’m a computer engineer (who was trained using modern techniques, at that). So this is all speculation. I think if you traveled back in time to the Roman Empire and told them how to manufacture this stuff, you could plausibly create a very modest computer. My main concern is powering the device, I still don’t know if that would work well enough. But there’s only one way to find out: Experiment…

Are you an expert at Roman history or the kind of engineer who remembers using a mercury delay line? I’d love to hear about other tricks they could have used!

Update: The comments on Hacker News make some interesting points about this article. And here are the comments on MetaFilter.

I participated in an eBay hackathon last weekend and you can see my entry here:

http://q.hscott.net/ebay

It uses Three.js and was hacked from one of the examples they have. I got the opportunity to play with a Leap Motion which was really cool, so if you have one, you can use it to move the shapes around on that page. I used the Javascript API to get data from the Leap Motion. It takes some practice, but you can pinch your fingers to grab a part of the mesh and move it around.Then let go and open you hand to stop moving it and reposition your hand. You can click on individual search results to take you to the page on eBay. There’s no way to click using the Leap Motion because I couldn’t figure out an effective way to do it using gestures. To navigate with the mouse, you can click and drag to rotate, scroll to zoom, and right click and drag to pan. The easiest way to look at the results is the grid view. You can zoom through the first panel of results to see the second. The sphere, line, and helix views are more there just because they look cool. Code for this is on my github.

I implemented a processor in VHDL and then pipelined it, adding handling for branch hazards and forwarding. This write up assumes knowledge of how pipelining and forwarding and branch prediction and stuff works. It would take too long to explain all of that in detail, so I’ll save that for a textbook. Maybe I’ll do a post on it sometime to explain the high level concepts.

First, here’s an example that doesn’t use any pipelining or forwarding. The code executing is:

The timing diagram looks like this:

The PC keeps track of which instruction is executing. At location 000, it is executing the first lw. You can see this in the write_data_out line, since it shows the address that it is fetching. It’s storing in register 2, which can be seen in the read_data_2_out line. Similarly, the next instruction at 004 (4 bytes, or 1 word in front of the last instruction), stores the word in register 3 (seen in read_data_2_out) and is fetching from AA. The data in those 2 locations is 0 and 1, respectively. The ALU shows that. The add instruction at 008 does 0-1, which is -1, or FFFFFFFF, as seen in ALU out. Next, we store that result in memory during instruction 00C (read_data_2_out shows FFFFFFF moving). At 010, 2 are compared and found to be not equal, so we go to 014 where we find that 1 are equal. Then the PC goes back to the beginning of the instructions, the first lw instruction. From there, the whole process repeats itself.

If you zoom in on the ALU result after the 3^rd instruction, you can see that it bounces around for a while before it stabilizes. This is because the way that the adder works results in the fact that different values will appear on the output while values are still sliding around and propagating through the hardware. Once it stabilizes, the value can be used. This needs to be taken into account when figuring out the maximum clock speed since taking the value too early will give you a wrong result.

Here’s a timing diagram from the pipelined processor I wrote:

Here, you can see that starting at the second clock, the PC is incremented by 4 each time. This is also reflected below by instruction_out. You can see that each clock brings in another instruction and fills the pipeline. Read data 1 and 2 both show the output from the decode stage. ALU result shows what is coming off of the ALU and into the memory stage. Write data out can be seen waiting for an instruction that requires a write to a register before going high. Branch and memwrite are both always low because there are no branches or store words in the given instructions. Regwrite goes high one clock before write data does. You can also see where the registers are written after each instruction that writes to them.

Most people like to drink coffee in the morning, but I don’t like coffee, so I drink tea. I wanted a simple device to help me make tea, but actual tea makers cost around $300. I don’t even need a full coffee maker to make tea, just a hotplate, so instead of Mr. Coffee, I built Mr. Tea.

One side has an adjustable hotplate, and the other side has a magnetic stirrer. The enclosure is an old PC power supply case.

The hotplate side was easy enough, I just tore out the guts from a hot plate I got from the store. The magnetic stirrer was just a PC fan with a hard drive magnet glued to the middle.

Turns out real magnetic stirrers for lab use are hundreds of dollars. To make the stirring pellet, I encased a small neodymium magnet inside some thermoplastic. Then I added a potentiometer to the side of the case and a power switch, so you could turn the fan on and off and adjust the speed. The hot plate is also adjustable. The main reason I wanted a stirrer was to keep from dirtying spoons. I put sugar or honey in my tea and I hated making a spoon dirty from just stirring. Plus it turns out that the magnetic stirrer works WAY better than doing it manually.

It’s kind of hard to see, but the water in this cup is being stirred by the magnet. There’s a little whirlpool in the middle.

Gibberellic acid is a plant growth hormone. Here’s what it looks like:

I got some from United Nuclear. Plants normally produce GA, and it’s what regulates their growth. So, if you introduce more to a plant, it will grow faster. There’s a ‘sweet spot’ of concentration you have to hit, or else the plant will die. It’s a very small concentration, no greater than 10 mg/L. GA stimulates the cells of germinating seeds to produce mRNA that codes for hydrolytic enzymes. That stimulates mitosis in leaves, and increases the speed that seeds germinate. It also causes plants to experience cell elongation, breaking and budding, seedless fruits, and can be used to break the dormant cycle. GA plays an important role at the beginning of a plant’s life as well. Before a seed has sprouted, it can’t photosynthesize, so it uses stored energy reserves in the form of starches inside the seed. GA signals hydrolysis by inducing the synthesis of an enzyme called α-amylase. That enzyme then hydrolyzes the starch into glucose, which the seed uses for cellular respiration.

I performed controlled experiments on 3 different plants: a tomato plant, a small cactus, and rose bushes. I had two of every plant, each in it’s own pot, and kept them physically next to each other during the experiment. The control was given only water, and the second plant was treated with a dilution of 5 mg/L. I took photographs of the plants with a reference ruler next to them over the course of several weeks. Here’s an example of one of those pictures:

And here are the final results:

The tomato plants died, because of the Florida summer heat.

That’s the rose bushes. My little brother is standing there for scale. The one on the left was treated with GA, the one of the right was not. There was some growth, but the plants didn’t start at the same height to begin with, so it looks more dramatic than it really is. But the one that was treated definitely grew faster than the control and grew about 2 inches more than the control did during the same time period.

The GA treated cacti grew about 0.5 inches more than the control did.